Use revive for kill with ulti or in the end game do a pentakill.

AP full staff is the great.
Buy the eleisha's miracle before the lvl 15.

Stop take the wall of pain before the lvl 13 no utility cost so bad lot of mana.
And if you don't take the wall you can deal more damages and if you'r ganked you can make two kill and they will stop gank you rest of the game.If they gank you again you have win because 4 or 5kill and with ulti you'r so much feed (they surrend you).

For AP in end game its the better.

Have good move in the lane and use revive in good time that its the more important if you play Karthus.Revive is the better spell for Karthus if you don't take with him you'r not the goodiest Karthus.