Basically you want to max your Q followed by your E then W, obviously putting points into your ultimate ability at levels 6,11 and 16. You may notice I have put a point into E at level 7, this is optional but I would recommend it as it really helps with the sustain in lane as it increases the mana returned for each minion kill.

PASSIVE - live forever (7 seconds forever)
The passive of karthus is very strong in my opinion, it means if teamfights if he can get in close to the enemy team he can do huge amounts of damage and they either kill him, which makes him essentially invulnerable for 7 seconds after his death where he can continue to deal lots of damage and cast spells for free and they cannot stop him. Or they ignore him and get destroyed anyway.

Q - Medium ranged small AoE damage, double dmg if on a single enemy (very short cooldown)
The Q is a very effective tool for farming in lane and singling out targets to burst down in a fight. Early game uses it to farm safely in lane from a distance and harass enemies who get too close. Try to utilise the full area of effect on the ability to last hit minions, if you cast the Q and just the edge of the circle hits the minion it will still deal full damage.

W - Medium/long range wall, shreds magic resist and has a stong slowing effect (long cooldown)
I max the W last as the utility of it is enough for the early game, it is mainly used in order to keep your enemies within your range when fighting and also to increase your damage output on them. It can also be a useful tool to prevent a gank on yourself as the slow is a strong deterrent for the enemy jungler to try and make a move under your turret. If the wall is well positioned it can slow and effect multiple enemies in a teamfight, ensuring you can pick up many kills/assists. Try to position it either on top of the enemy you want to kill or in a position where they must walk through it to get away.

E - AoE effect around yourself also provides a passive of mana returned for every kill (no cooldown)
Be aware that when the E is activated, the mana return passive is disabled, so you will not gain mana for kills earned with your E. The E is a very large damage effect that can be used to punish any enemies that come into melee range to try and kill you. Early game the E will drain your mana bar extremely quickly so I would recommend only using it if you have to, primarily try to use your Q to farm, however the E can be used to quickly clear a wave of minions if needed.

R - global damage to all enemy champions (very long cooldown)
The signature ability for karthus can be extremely strong through all stages of the game, in sync with his passive even after his death karhus can be a massive pain for the enemy team.
When you first hit level 6 look to top and botlane and see if there are any low enemy champions that could be killed, if not then hold onto it and try and always keep an eye on the minimap for any fights that could be won with the help of your ultimate. Be careful however, do not cast your ult in a position you can be easily reached as your ult can be cancelled and go on the full cooldown with any hard cc, such as stuns etc. Also despite your passive keeping you alive after death, if you die while casting your ultimate it WILL be cancelled. If you know you are going to die wait until AFTER you have died to cast your ult, this way you cannot be stopped as you are untargetable.