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.
With this build I usually get around 1,500 AP. I get Clarity for early game spamming/harrassing with Lay Waste. Ghost or Flash are both useful, but I usually perfer Ghost. You buy Mejai's first and start stacking from early game, and if you are good you will have near 20 stacks by mid game. Try to feed yourself early game. If that happens, its pretty much GG. All Karthus needs is a couple stacks to get a headstart from the beginning. Eventually, you sell your Mejai's for another Archangels Staff, and if you haven't won yet and still have money to spare, sell your boots for an Archangels too. This is a build for doing MASSIVE damage, not for surviving. It is alright if you die a lot, as long as you are taking enemies with you. And you need to be watching what is happening in other lanes and all over the map. ALWAYS be ready to use Requiem, and only use it if you are sure it will kill the enemy. You have an idea of how much damage it will do, but if you have time, check the enemies HP and Magic resistance to make sure you can take them. And don't worry about teammates getting angry at you for stealing kills. If they want to win, they will feed their Karthus. A fed Karthus is a happy Karthus :D.
Yes I do realize that this build sucks at pretty much everything other than mana and AP, but your massive AP will make up for your lack of magic resist and other things. If the enemy team has a lot of annoying melee characters that kill you too often, consider trading an Archangels Staff for some Thornmail.