Not Updated For Current Season
Lots of people will groan when they see a lack of survivability. Well, what if the idea here is NOT to survive? Karthus is unique, he can do some of his best work AFTER he dies. So put aside your fear of death, Karthus does not fear death, Karthus embraces it, and takes everyone with him.
Your core items are:
Mejai's Soulstealer - stacks to 180 AP
Hextech Revolver - 15% spellvamp (20% with Will of Ancients)
Rabadon's Deathcap - 150 AP + 30% of total AP
I'm not saying this build is the best, but it's ideal for my play style. I get different items earlier or later depending on how the games going (eg. Fighting noobs means I can get Mejai's Soulstealer first, and Hextech Revolver later - less need to heal from harassing, + earlier stacks from easy kills/assists).
At start, you badly need Mana, and some people will aim for a Tear of the Goddess by getting a Meki Pendant first. But I prefer the Doran's Ring; it's 4 mana regen instead of 7, but the extra 100 Health and 15AP really helps. And Karthus rakes in so much coin, it's not hard to get all the gear you want.
When you have 1200 gold (or even ideally 1550 gold), go buy either Hextech Revolver or Mejai's Soulstealer depending on if you need healing, or stacks early.
Depending on how much gold you have when you suicide-kill and find yourself at nexus, you may wish to grab some items earlier. Such as Will of the Ancients, before completing Rabadon's Deathcap (it's only 900g + Hextech Revolver, and offers 5% more spellvamp + AP)
Lich Bane gives you yet another nuke you can use every 3s. Like you need it... It also gives some other helpful bonuses, Mana, MR, Movement. You may want to grab Zonya's Hourglass or even Rylai's Crystal Scepter instead depending on how games going, and what you prefer.
Playstyle / Tactics
Karthus can auto-attack AND cast Lay Waste at the same time. So I right-click a creep to constantly attack, while hitting Q Lay Waste to harass enemies and last hit creeps. Auto-attacking just keeps the enemy guessing where you're attacking, as Lay Waste is really hard to see coming. You'll have noobs running in chaos, and even pro's will be dodging too much to last-hit for coin. Just be ready to run back if they come for you, turning only to Lay Waste in front of them.
Clarity will restock your mana as you need it, on top of last-hitting creeps with Defile's passive mana gain.
When you and your lane-partner go in for a gank, switch Defile on and throw up Wall of Pain behind them. If they run, run with them and drain them to death. If they fight, spam Lay Waste and watch them whither away. Ignore creeps, the more creeps there are, the more your Hextech Revolver will vamp the life from with your Defile. The more the better. WATCH YOUR MANA though, early game, Defile will drain it fast. So kill fast, or get back. Lay Waste is your best damage/mana ratio, spam it.
As soon as you have Hextech Revolver, you can grab golems blue buff. VERY useful for the huge mana regen, but also the reduced cooldown for faster Lay Waste spamming.
You can even jungle if you want to (though hardly a good use of Karthus' ample talents for farming and killing, even tanking).
Mid-late game, it's OK to suicide in team fights. Don't worry about loosing stacks on Mejai's Soulstealer, you'll get them back easily when you kill 2-3-4-5 enemies in one go. Again, throw up Defile, use Flash if you need to, and run in. If they try to run away (which they probably won't, they'll focus you.. fools!) throw Wall of Pain just behind them and run at them. If/when you die, keep Defile up and spam Lay Waste for more kills. Finish with Requiem if any escape low health. = 1 of you dead, and 2+ of them dead. and you're team pushes on unchallenged to take out towers/inhibitors.
Late game, which you'll never reach with Karthus, because he's so OP. Zonya's Hourglass will allow you to TANK for 2s in team fights, or under turrets for even more hair-pulling fun with enemies. Personally, I've almost never used it.. never needed it, i prefer dieing.
NOTE: With Defile toggled ON, you will not earn mana for kills and costs a lot of mana, but it does kill large creep waves fast. It's preferable to slaughter with 2-3 casts of Lay Waste on each 3-group.