My name is Michael and I am a Platinum 3 player on NA and my username is Main Got Nerfed. I love to play late game scaling/farming champions! My favorite class is duelist but for some reason when I played Karthus for the first time I fell in love! Literally all you have to do is farm and press R for kills to become the carry! Keep in mind I'm not a pro Karthus by any stretch of the imagination but I am pretty decent with him and this guide is to help you get there too. I'm going to keep this guide as up-to-date as possible.
Karthus was buffed on 8.21 :)

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Are you tired of enemies escaping with 1 hp? Are you tired of falling off late game and being useless? Do you like having the last laugh!? You've come to the right champion. Karthus is a champion unlike any other!

Pros--

• Very strong mid and late game!
• Strong team fighting!
• High damage output
• People always underestimate you
• Useful even after you die
• Still good even when behind
• Stacks tear insanely fast (If not THE fastest)
• Great waveclear with 2 AoE abilities

Cons--

• Weak laning phase
• Needs items to be useful
• lacks hard cc
• Mana hungry
• Literally no escape besides flash
• Damage is dependent on hitting Q's

After trying many combinations, flash and exhaust are my favorite. I've tried to run TP and it is good for the early game but in the mid to late game, you lose out on dueling potential without exhaust. However, for top lane I would recommend TP for overall map pressure. If your mid laner takes TP, then I would take exhaust because I love to duel. Ghost is ok, it can be viable some games but most of the time I would not recommend it. You need flash to avoid ganks and quickly reposition in fights. Ghost can be useless if they have a Jarvin on their team or huge displacement ults like Gragas. 99% of the time if you get Rylai's it's just as good as Ghost.

The Rod of Ages is a very strong item on Karthus as it gives him all the stats he needs: health, mana, and most importantly ability power. Build this item first in order to get the stacks as soon as possible.

The next item that you should try and build is the Archangel's Staff. This will give you loads of mana plus a juicy 80 ap. The real reason behind it is for when it transforms into Seraph's Embrace. The shield lets you bait enemies into fighting you. It's very useful in team fights when a predictable burst will hit you (I.E. a lux ult, or a Darius ult) so you can soak most of it up. If you are taking tower shots, use this item to negate a shot or two. It's personally my favorite item to get on a caster mage.

Rabadon's Deathcap is a self-explanatory item- pure power. Get this item 3rd if the enemy team is really squishy so you can kill carries as fast as possible.

Otherwise, Rylai's Crystal Scepter is the next best option as the slow lets you hit your Lay Waste much easier and once the enemy engages they won't leave until they are dead.

The perfect synergy with Rylai's is Liandry's Torment. Your E Defile will dish out a solid amount of magic damage even to tanks due to the % current HP passive on Liandry's- Which is twice as effective due to Rylai's!

Zhonya's is a great item against assassin's like Zed or Talon. If you fall at all behind against an assassin or a one shot mage like Syndra or even Lux, you should buy this item. This item gives you survivability and outplay potential if you use it correctly. Don't be discouraged if you don't know how to use it at first. It takes practice to truly use this item effectively.

Pretty straightforward item now- get this item if your opponents have heavy healing. ALWAYS take this item against Dr. Mundo and Vladimir. Otherwise, keep an eye on enemy's builds. If they are building items like Bloodthirster or Ravenous Hydra, consider building Morello's if those enemies are fed.

According to lolwiki, this item becomes gold efficient if your opponent has 90 magic resistance or more. If your opponents built items like Maw of Malmortius or Spirit Visage, I would buy this item. Don't make the mistake of buying this item if you opponents aren't building and magic resistance! It's better to build AP if they have no resistances. Some champions, like Jax can get magic resistance from abilities ( Grandmaster-At-Arms) so take that into account when calculating how much magic resistance your opponents really have.

Our main tree will be the Sorcery Tree, since we use abilities as our primary source of damage. For the keystone, I take Arcane Comet over Summon Aery or Phase Rush. Arcane Comet has the best AP scaling, and we don't need the movement speed from Phase Rush. We also cannot utilize the second aspect of Summon Aery which is increasing shields given to allies. Manaflow Band is pretty explanatory- we are very mana hungry and need every bit of mana we can get. Transcendence is far better than celerity due to celerity nerfs and I take Transcendence every game as Karthus. Finally, I take Gathering Storm because it matches my playstyle- waiting until late game for maximum strength. Scorch is also a viable option.

The second tree I chose inspiration. There are plenty of options in this tree, so I would encourage you to pick ones that suit your style- that's what these runes are all about. I personally choose Magical Footwear and Perfect Timing. Future's Market is another good option.


Another viable option is the precision tree. The new Presence of Mind helps with ultimate CDR and there is damage boosting runes at the bottom of the tree such as Last Stand and Coup de Grace (pronounced "Coo Deh Grass" if I am not mistaken)

Early game

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This is the laning phase. Your strategy for laning will change depending on the matchup. Is your opponent mobile, like Riven or Yasuo? Don't push or they will freeze the wave and leave you susceptible to ganks. Instead, match their push or simply farm under tower. Are you against a tank or immobile fighter? You should be focused on poking them with your Q Lay Waste. Look at the health of your own minions and you can predict when they will walk up to cs, then you can more reliably hit your Q. Do your best to make sure the minion is isolated, so you can deal double damage and ensure the last hit. Take into consideration of the slight delay in your Q or you might miss some CS. Once you hit 6 make sure to check on other lanes to see if you can ult for kills. Generally this is how you'll be getting all your kills early game. Your goal here is to CS as much as possible and die as little as you can. If you wish to push the lane (right before you recall, or after you kill the enemy laner) you should Q all 3 of the caster minions until they die then carefully last hit the melee minions.

Mid game

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At this point there might be small skirmishes and roaming champions everywhere but you still have not reached your maximum strength. Depending on how fed you are, you might be able to 1v1 the enemy ADC assuming you can hit a couple Qs. Still, focus on going to any lane with big waves to get CS. You can go for some kills at this part of the game but more often than not, you really should just be playing safely still until late game where your true power shines.

Late Game

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AKA you have 4-5 items.
You are a very powerful champion and better yet others have fallen off at this point. Your mission here is to dictate and win teamfights. You can use your Wall of Pain to catch an enemy out of position for an easy engage with the 70% slow it gives. Once they get in your Defile range, they will be dead. Know that a lot of your damage comes from your Defile so you want to be close enough in a fight to have your E ticking on enemies but far enough that you won't be assassinated.

Karthus is one of the best team fighting mages that exists. He has a high damage AOE ability ( Defile) and his Wall of Pain reduces magic resistance by 15% of all who walk through it! Meaning your teammates benefits from this too.

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If you find yourself setting up for a teamfight and both teams are in the poking phase before the fight begins, use your Lay Waste relentlessly as it has a low cost and cool down yet dishes out a lot of damage. Don't be afraid if you Lay Waste and the target isn't alone for the double damage. Any damage is good damage. Now the team fight begins. Your role is to stand back and take out the highest priority target in your range with your Defile and Lay Waste. If your target dies, repeat with the next highest priority. If you know you will die soon in a fight walk straight into the enemy team. Yup, that's right. Run right into them and give them pain! Your Defile will constantly be doing 100s of damage and once you die you can safely cast Requiem without being afraid that CC will interrupt it. Do not be afraid of death. That is the difficult part of Karthus sometimes. Be real with yourself and if you know there is little hope of an escape, go in and die to deal as much damage as you can so your teammates can finish off the fight. This is so important that I would even Flash into the enemy team to get their backline in the damage too. Remember, you are death.

There is an actually good reason for having no CDR in my main Karthus build. Short answer: Defile does not scale with CDR.
In depth answer: You may want CDR for your ultimate Requiem, but in reality, there are few times in a game where you can use your ultimate effectively. If you use your ultimate that seldom, it is clear that you don't need CDR to repeatedly use that ability. Also, Defile becomes a huge source of damage in the late game but it is a toggled ability so it doesn't scale at all with CDR. Finally, if even after that you still want to grab an item with some CDR, your purchase won't be 100% cost efficient. You are paying for that CDR when you won't even need it so therefore, you should stay away from CDR items in most situations.

WITH THAT BEING SAID-
After doing many mathematical calculations, I discovered that building CDR is viable. The main reason for doing this is to get Lay Waste cooldown lowered, so it can be spammed even faster. The results from my calculations are simple. If you can land your Qs, build CDR. If you aren't good at landing Qs, then this build which has more AP will be better. The exact cutoff is that if you can land 47% of your Qs or more, it's better to get CDR. Note that CDR builds have less AP overall so your ultimate won't be as strong as you may be used to. I recommend checking out Ding D-ng's guide for good CDR builds and playstyles.

TL;DR- If you're skilled, buy CDR.

-To charge tear of the goddess faster, have your E on while you recall and sit in base. Defile counts as a spell casted so it will passively charge while you recall/are sitting in base. However, this should not be used to stack tear outside of these situations; using Lay Waste is a much better method

-You can convert mana to health at an affordable cost if you have at least Tear of the Goddess and Catalyst of Aeons. If you keep Defile on, it costs mana which is converted to health, and part of the mana spent is refunded. You heal up to 25 health per second!

-You may deactivate your E Defile while recalling and your recall will not be cancelled.

I would not first-pick Karthus for top lane. He is easily countered by fighters with gap closers. I would only pick Karthus top if you know that you will be facing a tank (like Shen, or Malphite). Generally speaking, you pick top lane if you're looking for an uneventful farm lane which will make you stronger, sooner in the game. Don't be greedy with your ultimate. Use it to help with jungler ganks in other lanes. Since you are a ranged top laner, you can bully opponents early on and deny cs.

I would like to thank jhoijhoi for making this guide possible with her easy coding instructions. Check her out if you're looking to create a guide.
Thank you, everyone, for taking the time to read my guide!