As Karthus it is best to go mid as he is the AP carry. Using lay waste to last hit minions can be hard at first and makes Karthus mana hungry. The meki pendant helps a bit, but it is good to get at least one rank of defile to help with mana problems. It becomes easier to last hit minions as well as harass the enemy champion with lay waste as the rank increases. Since it does double damage when it only hits a single target it is good to harass while the enemy is coming in for their last hitting. lay waste's cooldown is really small (only 1 second!) but this attributes to the delay in cast and actual damage being dealt, so try to adjust for any movement the target may make. Try not to go back to the store unless it is absolutely necessary or you have enough money to buy a tear of the goddess.

While these items are what I always get they are not necessarily needed in that exact order, so I recommend getting the items you need most at any particular time. For example if your team isn't having much trouble with finishing kills and you don't need to use requiem as much, then postpone getting Morello's Evil Tome until late game. Blue buff works well with this for the added cooldown reduction and mana regen, and can be very helpful in teamfights where you keep defile on for the entire duration of a 10-15 second fight.