I find Karthus easily one of the ****tiest heroes in the entire game. His nuke is terrible- it is almost good against a single target, but not quite there. He's so incredibly lackluster that I can barely believe it. He provides no CC, no burst, no real damage except for his AoE drain which is his only redeeming ability. His ultimate is okay, but frankly, the fact that he has to channels it makes it a lot worse than it could be.
His only good ability is the wall that he can put up, which does a lot of things at the same time and is fantastic. That's the only good thing about him I dare say.
If you'll excuse me, I'll make parallels to the dotaheroes he's copying- zeus ultimate is instant, and can be used in the middle of teambattles and chase down. Zeus is also an amazing nuker, dishing out some of the best magical damage ever, which helps him plenty. He's also like leshrac with his AoE- but leshrac had two AoE spells around him, and no surprise there, since if you're going to be close to the enemy team you better kill them off quickly as hell or die trying.
I don't know what sort of role he fulfills, and I imagine the vague idea behind him was that he would harass and lasthit in his lane with his nuke- but lets face it, that nuke becomes completely worthless outside of lanes and even in lanes, some decent lasthitting and harassing will do the same job.