***Note-Please do not just vote this build down if it did not work for you or you feel like voting me down for some other reason. Leave a comment as to why you did not like it, and if you deem it necessary, offer a suggestion for improvement.

This build is utilizing Karthus for a nuke that specializes in farming minions early game and going for kills mid and late game.

For the skills, I recommend starting out with lay waste, primarily because it has such a quick cooldown. Getting that to level 5 as quickly as possible helps to farm minions and to do a decent amount of damage to enemy champions.
With that being said, however, you should at least get level 1 of defile and wall of pain before level 6. If you have a good team that is getting kills early game, then the wall of pain might be worth leveling up sooner. The skills build listed is for a player that is trying to farm minions early and go for kills later.

For the items, I highly recommend starting out with Archangels Staff. It gives you Mana and Ability Power 3% of your total mana. This build gives roughly 3800 Mana as the max. That converts to 114 AP.

The next item listed is Lich Bane, however, if you feel that you need boots, go with the sorcerers shoes or boots of mobility. If you are cautious enough with Karthus, the boots are optional. But be warned, if you are having to run a lot from enemies, you better go with the boots. But back to Lich Bane. 80 AP is nice, but how about turning your ordinary attack into one that does potentially at least 500 damage late game every 2 seconds? Makes Karthus a lot more deadly when combined with Lay Waste.

Rod of Ages. Obvious reasons. Health + Mana (yea archangels!!!) + AP. Gives player a little more health to work with to stay in fights a little longer.

Here is the place where you can start switching out items based on what your game gives you.

Deathfire Grasp - decent AP, Mana regen, and ability cooldown bonus. Oh, and it provides a pretty good 5th spell to hurl at people too.

Void Staff - AP plus magic penetration = Hurt enemies. Another archangel's staff may be appropriate here as well.

Zhonya's ring - AP with another spell.

If you have any suggestions or comments, feel free to do so. I will update the page as I get time to do so. Happy hunting!