Karthus is a mage who can inflict enormous damage over time and all over the map, which is compensated by a lack of mobility. AP and magical penetration are the stats you need.
- Sorcerer's Shoes is a good choice. Once again, any magical penetration available is good to take.
- Mercury's Treads is an other option if your match-up neutralizes you completely without you being able to fight back (a Syndra having the advantage for example).
- Rod of Ages will make you less vulnerable and increase your mana pool. Finish this item as soon as possible! If you're confident and don't need any extra HP, Luden's Echo is a great alternative.
- Morellonomicon gives you more AP and magical penetration. With R - Requiem, you can also cut the regeneration of an enemy on the other side of the map, useful if your allies are in the middle of a battle.
- Rabadon's Deathcap will bring even more AP to your build.
- Void Staff will give you even more magical penetration. This will increase your damage on tanks and inflict raw damage on the most fragile targets.
- Speaking of tanks, if the enemy's composition is full of them, Liandry's Torment can be very useful for you. In addition, W - Wall of Pain will increase the passive damage of the item if an enemy passes through it.
- Spellbinder is in the same spirit as Rabadon's Deathcap: to gain a lot of AP. Use its active at the right time to maximize your damage.
- If you're playing against an assassin, needless to say, Zhonya's Hourglass will be your best option. E - Defile continues to inflict damage around you while you are in Stasis.
Karthus is a character that you must play aggressively to win your match-up, but be careful not to get punished by the jungler. It is therefore essential to put a lot of wards around you in order to protect yourself from the opponent's ganks. The more vision you have, the more damage you will be able to inflict on your opponent.
Level 1: Karthus has one of the best level 1 in the game, take advantage of the low mana cost of Q - Lay Waste to do maximum damage. If you are precise enough with this spell which remains difficult to touch, you can completely zone your opponent and thus make him miss minions.
Level 2 : W - Wall of Pain allows you to slow down your enemy in order to cast your spells more easily. It is also your only tool to save yourself from a gank except your summoner spells: use it sparingly.
- Q - Lay Waste and W - Wall of Pain both give vision when used. Use them to check bushes where you don't have a ward.
- Once you've made a point in your E - Defile skill, you'll gain mana every time you kill an enemy unit or when you shoot down a turret.
- W - Wall of Pain reduces the magical resistance of enemies passing through it.
- When your passive is up use your ultimate spell soon enough or you won't have time. You have time to use at least three times Q - Lay Waste then R - Requiem