When you start:

Try to get mid. You must have your ultimate ready as soon as possible. Start by taking your boots, 2 healing pots and 1 mana pots. Don't be greedy! you have to stay alive to not lose your leveling advantage! Harass as possible with lay waste but never push to hard! Your goal is to have that precious lvl 6. Don't worry about the enemy turret. As soon has you got your ult verify other lane to see if the enemy champs are low life, if they are make your ult. that will give your team the time to destroy their tower and come help you with your's.

Mid game:

Get yourself mejai to be able to stack it as soon as possible. The more kill you'll make the more powerfull you'll get. Always be alert about gank. That's what make the difference bethween a killing karthus and a feeder. When you see someone trying to come gank you dont hesitate to put your wall of pain and use your defile while running for your turret.

Mid/late game: On team fight always try to not be the one in the front or in the back, stay with the group.

good tips:

When your being focused use your Zhonya to get invulnerable while using your defile. It always fun to see people bashing you and get killed without having lost a single hp.

Use your wall of pain behing enemie line to prevent them from fleeing giving your melee dps time to finish them.

ask a teamate to help you get bleu buff as often as possible. Defile just empty your mana pool to fast.

and with that its all you need to know to have a great match with Karthus and watch the oter team raging a you.