While most builds are mostly focused on Tahm Kench as a Devour babysitter, mine isn't.
As a support my Tahm Kench uses his great dueling strength to force fights early on. By mid -game Tahm Kench will use his Devour to gank or counter-gank, even his own lane exerting global map pressure.

With this build TK roams like Bard and fights like Udyr.

Sample of TK's early power solo'ing a Camille and Irelia while underleveled. Play Video
The domination runes are stronger early and also in small fights (Tahm Kench isn't great in 5v5's). The AS boost is so strong that many players don't properly react in time making it easy to stun or devour.
The precision runes are more balanced. There's less downtime and they are much stronger when fighting more than 1 enemy.

V9.2
Abyssal Voyage Abyssal Voyage
NEW EFFECT: Allies in combat with enemy champions can no longer jump into Abyssal Voyage.

V8.19
Stats
Base health reduced to 540 from 610.
Health growth increased to 102 from 95.
Devour Devour
BUG FIX: Allies Tahm Kench has devoured will no longer sometimes be targetable when first swallowed.

V8.16
Tongue Lash Tongue Lash
Slow duration reduced to 1.5 seconds from 2.
Cooldown reduced to 5 seconds from 6.
Devour Devour
Devour duration reduced to 4 seconds at all ranks from 4 / 4.5 / 5 / 5.5 / 6.
Enemy devour duration reduced to 2 seconds at all ranks from 2 / 2.25 / 2.5 / 2.75 / 3.
Cooldown increased to 28 / 25 / 22 / 19 / 16 seconds from 14 / 13 / 12 / 11 / 10.
NEW EFFECT: Refunds 50% of its cooldown if cast against an enemy.
Mana cost reduced to 60 from 90.

Pros: Very good duelist A lot of utility w/ strong skill set Devour is effectively a suppress so it denies a long list of spells and effects Stupidly strong early Strong counter against assassins Strong map pressure through Devour Hard to kill Strong against fighters

Cons: Very slow Poor poke Vulnerable to %hp damage and hyper-carries Vulnerable to crowd-control effects Reliant on stacks Weak when outnumbered Weak against kite Can get outscaled by hyper-carries and AoE fighters

Domination

Hail of Blades procs TK's passive 3 times for damage and enabling quick stun and devour. Hail of Blades is extremely strong for Tahm Kench. He's a great duelist and this rune makes him overbearing, especially in the early game when he can 1v1 most champions and even solo a lvl2 jungler while lvl1.

Taste of Blood improves laning as TK can get some health back on trades. Cheap Shot is an option, but currently it's damage is too low.

Good rune that provides extra map visibility. Synergy with supports which constantly ward and remove wards. Superior ward coverage wins games. Currently gives a damage bonus after clearing a ward which is good for roaming.

Relentless Hunter provides a lot of easy out-of-combat ms which is very important for roaming and backing up the whole team. Tahm Kench is very slow and this helps a lot.

Inspiration
These 2 runes provide a boost of hp/ms that improves TK's early fights and it helps to compensate for his health nerfs in season 8.
Biscuit should ideally be used at low health to bait (or survive).
I currently favor these as secondary.

Time Warp Tonic makes so consumables instantly restore 50% hp/mp, but also grant 5% ms. Good stats early and TK usually wants a Refillable Potion so this advantage keeps in the mid-game.

Biscuit Delivery grants consumables to heal for 15% of missing hp/mp which is really significant when TK is low as he has a significant pool. Also grants some max mp which isn't bad as TK usually doesn't want to buy MP.

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These 2 runes help TK to stick to his targets. Many champions without a lot of mobility/cc have trouble running from TK with these runes.

Approach Velocity provides a lot of mobility and is easily one of the strongest runes for Tahm Kench. Against enemies TK can now use Tongue Lash to slow and get a 15% speed bonus to close the distance and then quickly attack with Hail of Blades to set a stun or Devour. Also provides 15% speed to reach CC'ed teammates for easier Devour.

Magical Footwear gives delayed free boots. Tahm Kench doesn't need to rush boots anyway since the other mobility runes will cover him. The extra +10 movement really helps TK later as he is very easy to kite.

Other rune options

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Glacial Augment can be interesting against several high mobility champions because he's not likely to stack passive in a single burst. TK is stronger against most assassins and fighters with mobility, but much weaker against strong duelists that might want to 1v1. When playing this rune it's less efficient to build attack speed and better to go tanky. TK is somewhat weaker alone and is better placed in a peeling role. The synergy with Randuin's Omen and Turbo Chemtank isn't bad and should be considered against some compositions. Taking Bone Plating and Overgrowth as secondary will improve TK's survivability.

Guardian provides a %hp scaling shield and high MS boost which are both good on TK. However, this rune is much tougher to use offensively than the other keystones and limits TK's options. I consider this a better pick for champions that have more mobility, like Braum.

Flash is obvious. Using it to save someone can turn a fight around and prevent giving a bounty.

Ghost has more synergy with TK than Flash and less cooldown. It can be used in some lanes, but it's best picked if TK is strongly countering the enemy composition.

Exhaust allows Tahm Kench to duel even much stronger enemies and provides an extra protection. When using Approach Velocity Exhaust will also work to provide the 15% speed against the target freeing Tongue Lash. Exhaust is increasingly more valuable as the game progresses to help TK against hyper-carries and AoE burst.

Ignite is strong in lanes with a lot of killing potential. A must have in lanes with sharp engages/trades such as Pyke and Thresh. It's likely better than Exhaust in the early game, especially if TK doesn't have Approach Velocity.

Early Items

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Relic Shield's line provides good sustain, gold generation and also helps controlling the minion waves. It's by far the best option for TK. Always rush for Remnant of the Aspect because it reduces recharging time, health recovered and makes it easier to proc. Tips: Save one charge for the second minion wave to help getting to level 2 faster. Most gold comes from cannon minions while caster minions can be good to quickly recover health as they are easily smited. If there's an enemy cannon minion dying and you don't have a charge you might be able to Devour it to stall for the charge. Against ranged lanes Devour can be used with Tongue Lash to grab cannon minions from a distance so TK can safely get their gold (more rarely, can be used to push wave).

Refillable Potion is pretty good for it's cost. Sells for 60g later to free a slot. Good synergy with Time Warp Tonic.

Control Wards are very important. Do: Control important brushes and chokepoints that you need use to get a kill. Control important brushes and chokepoints to reduce gank threat or to escape. Control objectives you want to take/defend. Provide vision of contested objectives. The enemy is doing baron and your jungler NEEDS that vision. Place the ward. Use control wards to bait fights. It's one of the sweetest baits ever. Avoid: Wasting time and money with control wards on places you can't control. If the enemy is going to immediately fight for that spot and you can't fight. Placing Control Wards at random spots. Vision is much less important on the sides of the map, away from objectives.

Boots

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Mercury's Treads are important for TK because he's easily CC'ed.

Ninja Tabi are important for TK because they greatly reduce damage from basic attacks. Does not reduce damage from on-hit effects.

Mid-game

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As Tahm Kench has been nerfed. He more than ever needs defensive stats to remain a relevant frontline. I recommend stacking some HP early then building towards the most threatening damage type.
Armor is usually better to get first to deal with the enemy adc and improve tower diving and dragon/baron resistance.

Armor items

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Randuin's Omen has good stats and is effective against enemies that crit a lot such as Yasuo or Tryndamere. In s9, many adc's are building crit. Randuin's active should be considered. It's very strong at engaging enemies as it can be used through other targets such as minions. Many players don't expect this. Champions with mobility such as Ezreal (without Flash) can be chased if TK lands Tongue Lash with Approach Velocity as Randuin's can be used to refresh the slow.

Ohmwrecker can feel like a weird choice, but it has decent stats for TK with slightly less health than ideal, but decent regen to compensate. It's Raptor Cloak passive is very good on most champions and hard to expect as Zz'Rot Portal is hardly used anymore. The tower disabling active can have it's moments, especially with teams that have good dives. It resets tower's scaling damage and it can also disable both of Nexus towers.

Thornmail is good against enemies that heal a lot such as Darius and many late-game fighters and carries. Unlike Randuin's Omen which exists for survivability (active slow is ok), Thornmail's role is mainly to cut healing effects and enable enemies to be killed. I don't really like this item outside of full AD enemy compositions because of low-hp and utility.

Sunfire Cape can be purchased if the enemy team doesn't do enough physical damage to justify the other armor options. Sunfire Cape doesn't do much damage, but has decent synergy with Wit's End. It can be used to split push lanes in a 4-1 or 1-3-1 setup depending on the situation as Tahm Kench can usually Devour to regroup.

Magic Resistance items

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Spirit Visage has good stats and improves Thick Skin.

Adaptive Helm is usually very important when facing strong DoT's and AoE spell effects for enemies such as Brand Teemo Cassiopeia Katarina Fiddlesticks Rumble Anivia. Consider buying an Locket of the Iron Solari later.

Attack Speed items

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Attack Speed is very strong on Tahm Kench to stack his passive. It complements Hail of Blades for extended/multi-target fights.
It's easier to buy into AS if TK gets fed, the enemy team has mostly one damage type or is mostly single-target damage.
Often TK won't be able to afford any of these and will have to play closer to the team.

Wit's End is a cheap way to get attack speed. It also provides up to 70 MR. Wit's End also deal 42 magic damage on-hit and reduce enemies MR up to 30. These bonuses are all very good for Tahm Kench's kit. It's a strong choice if TK's team is strong on magic damage or the enemy team is heavy on AP bruisers. Not an ideal item for protection against heavy AoE damage such as control mages.

Stinger is a good attack speed option to deal with heavy AD fighter/duelist teams. It's cheap so it doesn't delay stacking health/armor. It can be built into Trinity Force late-game which provides a lot of useful stats.

Late-game

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Once Tahm Kench has enough to keep himself alive it's usually a good time to get some extra utility.

Utility

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Knight's Vow is a common support item. It's stats are average so I only consider this worthwhile if there's a fed carry/fighter that Deserves it. Good into the late-game when Tahm Kench can commit to peeling.

Locket of the iron solari is a good option against heavy AoE enemies. A good positioning is still necessary.

Righteous Glory is good to close the gap against enemies that kite a lot. Very situational item as Tahm Kench usually engages with Devour, but has it's moments. Experienced players will quickly adapt to avoid your charge so it needs to be timed.

Trinity Force has a lot of good stats. Besides the utility from extra speed and CDR, it gives nearly as much DPS as a Guinsoo's Rageblade. Too expensive so it's not a common purchase. When building it purchase order should be Stinger then Phage and lastly Sheen. It massively improves Tahm Kench's performance in duels.

Warmog's Armor is the item to deal with a lot of poke and magic damage in general. Usually TK can't afford this because it doesn't help against ADC's, especially Blade of the Ruined King + Guinsoo's Rageblade.

Frozen Mallet is good against some fighters with a lot of mobility and assassins that they can just run past Tahm Kench before he's able to stack the passive. Frozen Mallet holds them down so TK can peel better. Another option is to get Glacial Augment. Randuin's Omen is generally more useful and has better stats.

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Tahm Kench's passive. Solid damage bonus that increases with stacks up to triple. At maximum level it provides 5.25% hp/hit which is very strong.
Tahm Kench's relies on this passive to empower his offensive spells so stacking it is very important.

Skillshot with good range. Procs a strong slow which is useful for engaging and a stun when empowered which is extremely important in fights. Very similar, but unlike Blazing Stampede it doesn't have an internal cooldown for the same target so Tahm Kench can re-stun as often as his CD allows, up to every 3s with 40% CDR.

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One of the strongest skills in the game and a very hard one to use since it can do more harm than good.
Defensively it can be used to deny damage and combos against teammates. Damage effects continue to apply on allies devoured so it won't protect from bleed, Ignite and many spells such as Death Mark.
Offensively, since it's dealt as a suppress and it lasts up to 4 seconds it can deny a lot of enemy abilities and effects.

Devour can be cast for free while Tongue Lash is mid-air to grab a minion or monster. This isn't very good to poke, but it's good to grab cannon minions and very strong at stealing monsters. Tahm Kench can easily steal the rift Herald. This can be used to cancel enemy recalls.

Devour is treated a suppression so Tenacity doesn't apply and it's only broken by suppression breaking effects such as Quicksilver Sash, but not mikhail's crucible.

Devour does a lot of %maxhp damage which is great for finishing enemies off. regurgitate applies a 0.25s stun so a good combo is Tongue Lash+stun -> Devour+wait for CD -> regurgitate+ Tongue Lash+basic attack.

Devour is effective to:

• Hold people up to 4s. TK can still a decent fighter against melee enemies.
• Deny most grab and engaging CC such as Death Sentence, Bone Skewer, Dark Binding, Sleepy Trouble Bubble and Cease and Desist
• Deny follow-up combos on allied champions (but careful with AOE effects as it will disable TK as well)
• Protect against skillshots, AoE and multi-target spells such as Trueshot Barrage and Requiem
• Block basic attacks and projectile targeted spells such as Ace in the Hole and Unleashed Power
• Block turret shots. Will cause turret to switch targets.
• Cancel Hextech Ultimatum
• Cancel most dashing abilities mid-air such as Dragon Strike, Hookshot, Body Slam, Leap, Slice and Dice, Vault Breaker, Glacial Path. Slower dashes are easier, but all are possible as long as the ability isn't unstoppable like Hecarim or Malphite
• Outlast Kayle's Intervention, Kindred's Lamb's Respite and most of Tryndamere's Undying Rage
• Get rid of annoying pets such as Daisy! and Summon: Tibbers
  
  Additionally:
• Reposition an ally that's on the other side of a thin wall (must be adjacent to river/base walls).
• Wait out Exhaust from an ally
• Remove suppression (damage still applies)
• Can spit enemies into friendly abilities such as Yordle Snap Trap and Event Horizon
• Throw enemies into skillshots

Devour is not effective:

• Against Aatrox revive
• Against Zilean's Chronoshift as it lasts for 5s and it's usually a low-hp enemy so to outlast it and get a kill TK would need to wait at least half its duration.
• Against unstoppable abilities
• Against DoT effects and most ongoing abilities
• Can't devour Zac's passive

Devour is good with:

• Allies with little mobility
• Allies with bursty rotations that have a clear "get-out" pattern
• Allies that have follow-up CC or burst

Devour is not good with:

• Allies that have defensive abilities such as Kled, Aatrox, Master Yi, Tryndamere. Without verbal communication it can be hard to help them
• Allies with channeling abilites. Devour can't be used during friendly channels

Devour has some weird interactions such as:

• If a player is hit by an ability like Death Sentence or Sonic Wave and the champion reactivates it to dash, devouring that player will make Lee Sin/Thresh to dash towards Tahm Kench. This is very important against Thresh as it can be used as a counter-engage by displacing him instead.
• Devour while getting hit by Keeper's Verdict will carry Poppy.
• Devour while getting hit by Dragon's Rage will carry Lee Sin.
• Abilities, auras and effects that aren't interrupted by CC won't be canceled by devour.

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One-point-wonder that's responsible for Tahm Kench's being harder to burst. Avoid trading in lane before having this skill or while on CD.
Great at tower diving. Switch aggro with Devour.
It's usually best to use the active to mitigate damage. Not too soon that shield breaks too quickly, but not too late that the shield fades (3s) before being broken.
The shield is ineffective against abilities that trigger on current health like Death from Below and Fear Beyond Death so it's better to activate it earlier than usual.
TK can't activate the shield while disabled so it must be used early if there's a chance of being bursted.

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Very powerful and often underrated skill. It's similar to Destiny but has a far shorted cooldown, can transport an ally and can be used much more liberally in the mid-game as Tahm Kench can do much more than Twisted Fate when in a bad position.

It's main use is to support plays. At level 1 it's range isn't very good, but with good map vision and positioning TK can be ready to jump in for a gank, invade or counter-gank.
Tahm Kench can give very strong ganks in top-lane, but also can self-gank bot-lane as well. For that, if possible, Tahm Kench wants to spend most time post-6 around mid-lane or ghosting his jungler.
Thresh and Bard are considered good for ganking, but they aren't as good as 2 players falling right behind an over-extended lane.

Tahm Kench needs to be careful to not use Devour too deep so he gets killed or too near AoE CC so he gets himself and an ally killed and neither too far so he can't support the fight in time.

It's really important for Tahm Kench to have solid defensive items to support his ultimate as often there's not enough time to get to an ally to help someone.

Abyssal Voyage is also good to get faster to a lane or objective and also for backdoor/split-pushing with the proper items.

While TK is very burly and strong he's also slow and not a real tank (think Garen, Galio, Rammus).

Having another tank in the team helps a lot into reaching that critical health/cc mass that simply shoves through front-lines.

TK's Devour works very well with friendlies that aren't completely squishy so he has time to close-in and activate and even more, time to Devour to assist.

Friendlies that have means to dodge or prevent death aren't ideal because it's much harder to tell when they need Devour or not.

Having a real engaging tool is better because Devour can't be relied upon. It's an easy way to deliver 2 easy kills into a pickoff if miscalculated. It's much easier to use when the enemy is already broken than trying to find and angle into a 5v5.

Invade

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If getting invaded, try fighting if possible. Tahm Kench has a strong 1v1 with damage and CC by Tongue Lash. He can hold himself against some 1v2's.

Good fights require vision (ward).

Enemy Jungler

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Tahm Kench can 1v1 almost any enemy jungler that's not too far ahead and doesn't CC/kite hard ( Taliyah).
If the enemy jungler is invading or doing the rift herald it's often a good choice to go after him, even if he has a level advantage as there's very little risk of dying with Exhaust/ Ignite.

To do so, have river or blue/red warded around 1:20 (depends where your jungler started). Try getting your lane slightly pushed to get a little priority. Tell the adc that you are leaving so he holds the enemy lane as long as possible and for mid to also hold his.

If Tahm Kench is level 2 it's possible to kill the lvl3 enemy jungler. If at lvl 1, it's only possible to stall for help of prevent a buff from being stolen.

Bot-lane


Bot-lane difficulty is usually based on the enemy support. Some enemy adc's do make everything much worse, but plays are mostly dependent on the support.

Generally it's best to avoid committing to fights unless the enemy makes a mistake like missing an important ability or gets in a very poor position. With a bursty adc, TK might do bolder Flash attacks on someone with 2-3 stacks to get a quick Devour into Ignite and your ADC.

[*] Very aggressive all-in types such as Leona and Alistar will usually result in an early fight which is usually favorable. TK has to decide whether to go for the enemy ADC or protect his own.
Defending the ADC is usually best if your ADC has no chance of survival or there's a lot of minions.
Attacking the enemy ADC can work well if he's in a bad position and your ADC can kite the enemy support or your jungler is getting close.

[*] Enchantresses usually make it hard to win a straight 2v2 unless you have a very good adc. They will usually try poking as much as possible. Early on there's not many options but to return some damage with Tongue Lash so your ADC can farm.
From level 5 and above Tongue Lash does enough damage to trade and Tahm Kench should have enough health that enchantress poke isn't too bad.
The major threat is getting ganked. Also, TK's biggest advantage is bringing in the jungler in with Devour.
Going in solo behind the enemy can work if the enemy adc is much weaker than yours.

[*] Mages like Brand and Zyra are terrible. It's very hard to close in and they do a lot of damage. Tahm can't do much without a gank. It's often good to roam so the enemy support is pressured into playing aggressive. Neither of them are very safe so it brings in opportunities to fight back.

[*] Long range mages like Xerath and Vel'Koz are annoying and will usually force an early recall. They are very vulnerable to ganks and Devour so just play it safe or roam.

Roaming:


Tahm Kench isn't very strong at forcing aggressive early plays as he has low mobility without his ultimate.
Sometimes he can't get kills at bot-lane, even with a fed adc, because the enemy will just play safe.
Sometimes he can't do more than proc Relic Shield because the enemy has too much poke.
It may be a good to roam.

I consider early roaming a stalling move. TK will forfeit pressure at bot-lane (which he may not offer at all) to pressure somewhere else.
TK wants to have map control and, if possible, fight other melee champions.
Ideally the enemy bot-lane will go more aggressive and, with help from the jungler or mid, you will be able to self-gank bot and not lose too many turret platings.

Top-lane
Usually TK support can win a fight against an enemy AD top-laner so it's nice if there's an opportunity to go there.
I usually don't bother unless the lane is in the middle or pushed towards the friendly tower and enemy top/jg isn't too fed. Fighting an average jungler is good.

Top-lane is very far so it's often best to wait for lvl6 if your adc doesn't have a strong advantage.

Mid-lane
It's not easy to gank mid with TK, but it's much easier to counter-gank.
Counter-gank 2v2's are usually good for TK because the bulk of abilities and spells will usually be used against the laner, at a minimum it's hard to die.

It's important to keep checking river bushes to remove wards so the enemy team can't control the river.

Once TK hits lvl6 his ultimate allows many options for team-play.
TK wants to gank every lane that pushes toward friendly towers, predict and counter-gank, and above-all try making plays with his jungler.

It's best if TK can stay near mid so his ultimate has more coverage.
I prefer playing with whomever is strongest or against the weakest, most vulnerable enemy.

How much roaming is done depends on how independent is your ADC. Some can't be alone so TK needs to stay bot-side.

Important:
As strong as Devour is. It's much stronger at throwing the game. Avoid doing very bold plays with little vision and early turret dives.
Don't bring a friendly with Devour into range of an AOE CC such as Lux or Zyra. It's always bad.

TK has 2 options for late-game:
[*] Charging enemy fighters so they are busy fighting TK.
[*] Peel.

Usually I'll play more aggressively if fed or in a favorable match-up. Having an attack speed item is essential. It's often easier to duel someone than peeling if the enemy champion can use it's mobility against multiple allies ( Irelia, Yasuo Master Yi Camille).

Tahm Kench peeling is strong against single-target abilities. It's a strong option if your team is fed or the enemy team has a lot of assassins/single-target burst.
Peeling is also a fall-back strategy if the enemy team has a lot of CC/AoE because it will be harder for Tahm Kench to engage.

When going aggressive. Fights are much easier, especially 1v1's against fighters because all their AOE and mobility are mostly useless.

It's important to be tanky enough to survive, but also to have enough AS to empower abilities and utility to protect allies.

Devour is still useful late-game. It can be used to quickly switch lanes to destroy building and fight objectives. Be careful when using it to engage because it's basically overextending, but it's a great way to push a small advantage (say 1 kill or low hp) into an ace.
Ultimate can be used to try to fight a started Baron. An option, if no better positioning is available, is to ult behind Baron so his model obscures TK's hitbox making it annoying for the enemy. TK will just annoy/distract while the rest of the team moves around. Exhaust can be used against the enemy jungler to pin him down .