Karthus is one of the best late game ap carries in Lol. He deals a ton of damage in teamfights and can help win teamfights.

Pros:
Great Farming ability
Global ult (nobody can escape)
One of the best passive (able to deal damage even after you die)
Nice slowing ability

Cons:
Easily countered

All you need to do in early game is farm. Farm and farm and go back when you can buy your catalyst. Harass your enemy with some q's but prioritize farming over harrasing. Try not to get harrased by your enemy laner too many times and be wary of ganks.

By mid game you must have a lot of farm and be finished your boots and RoA. You must be building your deathcap now. In mid game try to farm for your items but at the same time help your team obtain objectives and help them out in teamfights and taking down towers.

This is where Karthus shines. Hopefully you almost have your build finished and have a few kills. Most teamfights happen in late game. Make sure you always have the blue buff for teamfights. Make sure to tell your team to ward baron and other important places and also always stick with your team. If you want to split push make sure you have teleport off cooldown. But your team needs you during teamfights as you do the most damage in teamfights.

When a teamfight starts, put your wall of pain in a place that will hit most of the enemy champions. When you place your wall, press e and walk into the middle of teamfights or flash into the middle of the teamfights. Don't be afraid to die, you still do damage even after you're dead. And once you die, press r immediately and watch the magic happen. If there are still anymore enemies alive, place q and try to help your team. With your full build and if your wall of pain hits every champion you should be doing close to true damage to the enemies even if they have some magic resist items.

Enjoy winning with Karthus and don't be afraid to die. Try to tell your team to force teamfights on top of your e for the most maximum damage.