Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Here is my Karthus build, played by me "The Big Box."
Now, some of these things I've put here may make you go "huh?" Such as revive, and the utility runes. Though I found with this odd build on Karthus, it can be very effective if you know how to play him.
PLAYING KARTHUS IN A GENERAL SENSE
Playing Karthus is somewhat difficult, and people complain about you pressing "r" all game long to get any kills. Though his other abilities are very deadly if used right, and with this build, you can go into the thick of things with your team even as a squishy caster. In my experience, Karthus's AOE mid-late game is very essential to help DPS down the enemy team, along with throwing lay wastes everywhere.
One on one, if you feel you have to take someone on or want to be bold (Since some like to be agressive or conservative in their play style).. it's always good to toss up a wall first, not just to slow them, but to lower their resistances while you pummel them with your AOE and lay waste. A single rank in wall beginning of the game also helps with chasing and helping out teammates escape. It's also good if your team wants to tower dive too, as it'll slow their escape and has a relatively wide effect if you place it in the correct spot.
Karthus's passive is pretty much based around this build. If you're brave and dont fear death, then your team likely wont either.
Lay waste - I max this out first, as it deals good amounts of damage, especially on one on one situations. It's tricky to time where to put the delayed blast sometimes, but other times, like with a few melee heroes, they stand in one spot for you to plant lay wastes under their feet, which is just devastatingly effective. Even if you die, you can likely take them with you in a trade off.
Defile - I max this out second. Very nice to have to add extra DPS if you're having a hard time landing lay wastes while chasing someone. Float along side them and continue to do damage. Though take care, as your mana may run out quickly. If the enemy team is all nice and bunched up, meander on in with this up and help out your team if they're in trouble. Again, even if you die, you can still help out for 8 seconds due to your passive, and you'll be up soon again thanks to runes or revive/teleport to get back in there.
Wall of Pain - A good utility spell. Good for escapes, chases and assisting team mates in either of those. Also good for bunched up enemies to lower their resistances and make them panic because they're slowed.
Requiem (Ultimate) - Many a Karthus I've seen loves to use this whenever it's up. Yes it's a delightfully fun ability to use, but there's actually skill involved with it. If the other team isn't hugely fed, this will help your team score aces, and allow you to finish off those pesky runners. People say "use it at the beginning of the fight".. I'd say no, bad idea. Use it when the enemy team is at low health, at least below half. At the start of the game, only use it for enemies at extremely low health and have -escaped- your other team mates. Dont use it to KS, that's just stupid and wastes the ult for another enemy that could be at low health and escape because your ult was on cooldown. Conserve your ult until the enemy is weakened, at least that's what I do. I save it mostly for runners or if the low health enemy gets away via ghost or whatever, when my other teammates cant catch them. With revive, after I've died from a team fight, I use revive, and pick off the runners from my ult if the team fight left everyone near death, so I dont have to wait for the respawn timer... since sometimes, you absolutely -need- to cast your ult right away.
Why I chose these runes
-% time spent dead runes help keep you waiting less when you die, it stacks with Karthus's passive, and stacks with the masteries buff too
Ability power runes - For that much needed umf early game. To help you deal out more damage, especially with your ultimate.
Standard ability power items. Zhonya's ring is especially important on Karthus though, so -always- get that no matter what, it works brilliantly with his Defile ability. I take rod of ages first for more durability, then depending on the team set up they have, I either take banshee's veil or Abyssal specter late game for extra mana pen. Some folks like to stack archangel's staff on Karthus, which works fine, dont get me wrong. I like the extra magic pen myself, that's just personal preference. Banshee's veil is situational... like with an enemy team who has mostly stuns.
Karthus is -excellent- with defending a tower, even by himself. If you feel that two, three, or maybe four enemies are about to tower dive you.. dont try to escape, because most of the time, you wont. Instead, toss up your aoe, then your wall when they are close to slow them so tower hits whoever hit you first. If you die, you can still cast your spells for free for 8 seconds. Most of the time (If the enemy team isn't hugely fed thanks to your team mates), you will hurt them enough that they at the very least back off from the tower and fail to destroy it. I cant tell you how many double/triple kills I got because they thought they could tower dive one lonely Karthus. It usually ended with me dying, but them all dying due to wall, aoe, and then ult. Sometimes other teammates come in to finish them off as well, such as pantheon or twisted fate, or gangplank's ult, or anyone with teleport. Thanks to your runes and passive, you'll be back up before they will.
So you may be put off by the revive and such, which is understandable. But it is a good spell for this build, and helps keep you active in team fights even after you die. Has it always met with 100% success in team fights? No, it hasn't. But most of the time it will, and it will basically guarantee you a few kills if you place it right.