Karthus is unlike any other mage, he does not really on epic burst damage, but a constant stream of damage that qualifies him as a DPS. Karthus has the ability to achieve triple, quadra, and often penta kills with or without his ultimate. He can achieve epic aoe DPS has one of the best slows in the game. Once you learn how to play with Karthus you can easily dominate games.

Quints - The flat health quints give you survivability and help you lane longer
Marks - Spell pen greatly increases you damage out put at early levels, and in conjunction with items can make a big difference late game
Glyphs - MR per level gives you a huge MR boost allowing you to survive better without a decent base MR
Seals - even when you have 2500+ mana M/p second is always helpful for Karthus' massive stream of spells, and early allows you to lane much longer

I go 9/0/21 nothing special, it is pretty standard caster setup

Starting out with a Meki Pendant and two pots should let you last long enough in lane to have 1585 gold when you port back, which will allow you to get a Mejais and boots. Before you upgrade your boots build a tear of the goddess. Once you upgrade your boots rush a Rabadon's deathcap, with 5 stacks and rabadons you will already have 300+ AP. If the other team is balanced or caster heavy build an abyssal scepter, if it is DPS heavy build an hourglass. Build the other one unless the other team has zero DPS/AP then build a void staff if 2+ people are stacking MR or skip this and proceed to build an Archangels staff. If you only bought one of the defensive items and did not buy void staff purchase another Rabadons. Late game do not resort to stacking Archangels because the AP increase is not worth the loss in survivability.

r>q>e>w
Always max your q first because it allows you to deal a constant amount of decent aoe damage that has a higher DPS than any other spell in the game. After that you E allows you to get mana back for kills and it is a strong AOE toggle spell with a good amount of damage. Your W is a strong slow and resistance reducer that is good for both escaping ganks and ganking, but you only need one rank early game. Your ultimate is your signature move, use it to kill all stragglers. Never use your ultimate to KS or deal damage in a teamfight, only use it when it is a guaranteed kill on champions your team cannot catch. Your passive allows you to ensure kills and change the tide of battles.

Ignite - Allows you to ensure kills and is great for countering healing champions (MUNDO) or ensuring a kill on someone invulnerable (Tryndamere).
Ghost - Allows you to escape, and occasionally chase (if you are chasing you probably missed your wall)

Early game - last hit with your q and your autoattack
Mid game - once you have your rabadons use your q to destroy creep waves
Late game - turn on you E press Q a couple of times and destroy the wave in under a second

Karthus is a great farmer, you should always be mid and you should always be one of the game leaders in minion kills

Generally during teamfights you want to stay back and spam your Q, get people in range of your E, use your wall to prevent people from escaping, and ult any stragglers.

If you feel like you are losing the fight, but you think you can get several kills if you could unleash yourself, then you must suicide bomb. Run into the middle of the fight spam all of your skills and when you die continue to do so. If you are doing this tactic it is preferable to get kills after you die so you can rebuild your mejais stacks.