Karthus is a hard scaling, immobile mage with the ability to kill after death. He has a global presence thanks to his ultimate, Requiem, which allows him to do damage to everyone on the enemy team, finishing off low health targets, or getting some early damage in before a teamfight. He has the ability to do a lot of AOE damage by simply standing near enemies with his Defile active and has soft CC and/or zoning capability with his Wall of Pain. These things make him a little more well rounded then some might expect
Passive – Death Defied
Upon dying, Karthus enters a spirit form that allows him to continue casting spells for 7 seconds.
Q – Lay Waste
Creates a delayed blast at Karthus' cursor position dealing magic damage to each nearby enemy. If the blast hits only a single unit it deals double damage.
W – Wall of Pain
Karthus creates a passable screen of leeching energy. Any enemy units that walk through the screen have their Movement Speed and Magic Resist reduced for a period.
E – Defile
Karthus passively steals energy from his victims, gaining Mana on each kill. Alternatively, Karthus can surround himself in the souls of his prey, dealing damage to nearby enemies, but quickly draining his own Mana
R – Requiem
After channeling for 3 seconds, Karthus deals magic damage to all enemy champions (regardless of distance).
A lot of Karthus usefulness comes from his ability to hit enemy champions with his ultimate, Requiem. Work to negate his ultimate with any number of shields, heals or stasis effects.
Karthus is pretty weak early, thanks to his lackluster bases. You can bully him early as a result, and even snowball yourself, if you play a volatile champion such as Zed, Leblanc or Syndra.
Requiem is a channel, which means it can be canceled. Work to keep Karthus in your sight, and try and use your hard CC to cancel his ultimate. If you don't have hard CC, just get as much damage on him as possible, it's likely that you can kill him during the channel, and still interrupt the ability
Karthus auto attacks are...Bad. He is reliant on his Q to farm and to harass people who aren't close to him. Use this to punish him, if you have mobility. Weave in and out of defile range to trade with him, and move unpredictably to avoid the Lay Waste on the way out. You'll outdamage him this way until he begins to reach the peak of his power.
Azir Outranges Karthus and prevents him from using Defile in teamfights by zoning him off with his Emperors Divide. That means that both the laning phase and post-laning phase are just a bad time for the Deathsinger
More often a top laner these days, he can still be played Mid and when he's against Karthus, it's a bad time. His mobility makes landing the Q difficult, and makes it easy for Yasuo to dodge out of the Defile after he does his damage. His flow shield can also mitigate the damage from Karthus' ultimate every now and then.
Shunpo counters his ability to outrange her, and she's mobile enough to dodge the Lay Waste. He can't stop her ultimate because he has no hard CC, and if he plays too passively, she'll go roam somewhere else, and kill somebody, snowballing anyway
If Leblanc lands an E ever, Karthus will die. She can get in, and get out, totally avoiding his Lay Waste after doing damage, and has the CC to force him to take her damage if she lands the chain. She has a strong early game, and can snowball hard by taking advantage of his weak one.
Banshee's is a must. It counters his ultimate, other sources of Magic Resist are also important. Mecury Treads are a good buy, thanks to their ability to lessen how punishing the Rylais he'll eventually buy will be. Face of the Mountain and Locket of the Iron Solari have a lot of value, since every bit of shield lowers the chance that his ultimate kills someone, and Locket giving AOE MRes is invaluable