Karthus's Skill Max Order: Q → E → W
Level 1: Conduct the Quinn Cheese level 1 and bully him as hard as you can. Try to side step his Qs, which are quite easy to do since his cast time is delayed. Do unpredictable movements as you weave in autos, and eventually karthus will run away when he loses the trade super hard.
Level 2-5: There is no fear in this matchup. His W slow makes landing his Qs easy, so if he lands W on you, instantly vault him to slow him back as you escape his W slow range. His W slows for 4 seconds, so after it wears off then go back in for a trade. If Karthus ever uses W in lane, do not let him breathe for the next 15 seconds. If you get into E range, vault immediately and run him down because he needs his W for disengage. Without his W, level 3 onwards you can run him down and kill him with ease.
Level 6+: The matchup stays the same, although make sure to not be low when you kill him because his R can return kill you. With nullifying orb, though, this is a direct counter to Karthus.
This matchup is free top lane.
Goal is to bait karthus W, and if he ever uses it, he can't play the game for 15 seconds.
Ideally, freeze against karthus, which will force him to over extend.
If you freeze, he has to be far away from his tower, so you can E him when he walks up and run him down until he flashes to his tower.
You must play close range to him because he out ranges you.
Karthus only wins if you play far back, so always be in E range of him if you can.
Use Q on him during trades, which makes it almost impossible for him to land his Qs on you while blinded.