Karthus is one of the best characters in my opinion. The way I try to play him is very aggressive in order to get many kills and assists. This means always saving my ultimate until champion fights. If I die in the process of trying to kill people but still do a ton of damage before you do, that is the perfect time to pop your ultimate and get many kills.

The items here are the ones I use in the match in order to deal a lot of damage. I try to get Archangel as fast as I can for the mana and ability power. This staff makes a big difference in early game. After that I start to work on Will of the Ancients in order to get spell vamp which really helps if you are getting a lot of damage dealt to you. The rest of the items are really there for you to get magic penetration.