Illaoi is a powerful but immobile juggernaut; she is meant to keep the enemy team in line and deal heavy damage to those who get too close. She has a lack of gap closers, but makes up for it by being able to attack an enemy wherever they are with Test of Spirit.

Pros:
Strong in team-fights
High damage
Great sustain
Able to distract opponent
Strong mid-late game

Cons:
Poor early-game
No gap-closers
1 misplay can get you killed easily

Passive - Prophet of an Elder God: spawns tentacles on walls that will heal you when they attack an enemy(comes from your q passive.)

Q - Tentacle Smash: this is your main damage dealing attack in my opinion, it has decent range and a powerful ad scaling, use this in combination with your E to deal damage to the enemy champion and their soul simultaneously.

W - Harsh Lesson: Your W is key for fighting enemies near tentacles and keeping up with them, your W allows you to jump your enemy and have your tentacles attack them once, use this constantly with your ult to deal massive damage and heal.

E - Test of Spirit: This is your skillshot, when it hits an enemy champion their soul is brought to you and stays there for 10 seconds. A percentage of the damage you and your allies deal to this soul damages your enemy. If you and your allies kill the soul or the enemy goes too far away from it they become a vessel, this means that they will spawn tentacles around them every 10 seconds for 1minute(6 times) that will attempt to attack the vessel, pre-existing tentacles will also attack the vessel. This ability will not directly kill your opponent here's how it can though:

1. Your opponent is trying to recall or runaway, a tentacle spawns, and attacks the vessel.

2. You take their soul while they have relatively high health and you damage the champion and soul simultaneously with your abilities, lowering their health while doing bonus damage.

During a fight, don't focus on the soul, focus the champion, if they start running away, go ahead and finish off their soul so your tentacles will do the rest of the work.

TIP: If you want to know where the enemy jungled is, E them, kill their soul or let them leave the area, this way your tentacles will act as wards, while annoying the enemy jungle, and if you're lucky it'll get you a free jungle monster, buff, or even kill the enemy jungler while they're farming.

R - Leap of Faith: Your ult is powerful, but can also be easily wasted. You jump up and slam back down dealing damage and spawns 1 tentacle per enemy champion in range. It also sets your W to a 2 second cooldown.

This is an ability you want to be careful with, and focus on the setup and execution of it.

Set-Up: Try to have your opponent at around half health(this is primarily for Laning phase) and ensure they aren't able to leave the ult range before you can land it. If using your ult to engage or turn-around a teamfight make sure your team will engage with you and not run-away. You also will want to get your E off before hand or as soon as you land your ult so you can deal some bonus damage when your Q and W + tentacles hit your opponent and their soul.

Execution: Once you've setup your ult, it's time to kill! While you're ult is active keep laying down your W, this way your tentacles will continuously lay down heavy damage and heal you. You'll probably end up in the middle of the enemy team so try to avoid as much cc as you can and stick with your team.

*This is the route I take but I haven't found the best one yet*

Start Gromp(smite it) then go over to Blue Buff, if you aren't confident you can kill it easily then wait until smite is on around an 18sec cooldown. Use Harsh Lesson whenever it comes up to get enough damage in. You can then go to Wolves or back and go to Red Buff.

Illaoi is weaker than most pre-6 so try to poke with your q during this time and be wise about your engages.
If you're beginning to lose a fight, use your w while near one of your tentacles to heal a bit and deal some damage.
If you're having a hard time in lane or just want to do some damage, try to use your e on your opponent while under turret to get in some free damage and force them to be watching out for tentacles.
When you plan on ulting try to save your w, your ult will put your w on a 2 sec cooldown but any cc during that 2 seconds can prevent you from dishing out all that damage. Thus if you save your w you can use it right when you ult and down come the tentacles.
Remember to get an e in during your ult so you can get in more damage and lower the chance that your enemy will get away.

Your ult is very strong in teamfights, but don't overestimate your damage and healing, you should make sure your team is ready to engage before or with you before jumping in and ulting.
Only use your ult when all 5 of the enemy team are within range so you can get 5 tentacles out and start dishing out damage, remember to focus their main damage dealer with your w so you can sway the teamfight in your favor very quickly.
If the all-out fight hasn't started yet or the enemy team is staying back a bit try to e one of them so your team can deal damage from afar and they will start spawning tentacles that will give your opponents grief.
REMEMBER: you're primarily a tank protect your primary damage dealer!