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.
start with a 200mana cristal and 2 health potions.
Whenever you can: go mid. This will help you and your team. You get your ultimate very fast and this can provide some early kills. It is very important to know what your enemy is doing. Lay some bombs and look what your enemy is doing. Is he running away, or running sidewards, or running towards you? Is he fighting minions isolated(lay waste deals double damage to single target!)or always among the minions. To get in the head of your opponent will help you to make farming etc imposible fot the enemy. When you reach lvl 6 you should have a fair amount of cash. Try to farm untill you can buy at least sorcerer shoes, or shoes and tear of the godess.
When your enemy has become low on health you can also go for a kill. Let the enemy come a bit towards your turret and use wall of pain, lay waste, normal attack, lay waste. Once he's slowed it should be easy to time your bombs. The make sure you have the kill use defile, but be aware it costs a lot of mana. (flash can be used to line up a final lay waste).
Mid game(lvl 10-16)
You should have the sorcerer's shoes (and tear of the godess). And be on your way to either lich bane or archangels staff. You might want to rush for lichbane because the effect of tear of the goddess still works when you buy archangels staff. But its perfectly fine to work on both lich bane and archangles staff. If you need AP fast, rush archangles's staff, if you need magic resist, buy magic resist for lich bane and AP for archangel's. It al comes down to the situation you're in.
Now it's ganking time. Keep farming and once your enemy drops below a certain amount of health (yours to decide) call in a teammate to go for the kill. Use wall of pain to slow and finish of with defile ON and lay waste. Once he slipps away with minimum health use Karthus' ultimate(R). Farm untill you can buy the archmage's staff. *I didn't go for the rod of ages because every Karthus player should try to play with minimum health and max mana. You can alway go for rod of ages to have a little more survivability, but I dont recommend it. Archmages'staff also grants a major AP bonus, which you will need! To give an example of the bonus. Once you have 2000 mana you get an additional 50AP!*. With Karthus and his spammable, low mana cost (Q) skill its easy to get the +1000 bonus from tear of the goddess/archangels staff.
Late game(lvl 16+)
You should be on your way completing your core item built.
Now the teamfights kick in for real! Be sure to not be in the frontline of battle. Stay back and spam lay waste. Then use wall of pain to slow and help yourself and your teammates get kills. Once an enemy slipps away make sure his hp is low enough to use R. Don't waste your ultimate, but also don't wait too long. Make sure if you get into the battle trying to kill enemys with defile on, that at least one tank should be with you. Firstly try to PAIN them with wall of pain.
The best thing is to let your team know that your ult is the initiation: after it hits you will engage. Now its clear to your whole team when to attack. If you use it too soon the enemy team will regen (most teams have some kind of healer), but if you use it too late (after initiation) you're channeling for 3 seconds and not dishing out any damage.
Another option is when you get too low to keep fighting savely, Then you retreat and hit your ulti at a save spot. You'll still be helping out your team without dying.