Karthus is one of the strongest casters in the league and the aiming point for gruesome hate comments. Master him and IP farming is a blast.

I'm going to spent almost zero words on this guide. Its just the setup to achieve marvelous things.

Marks: magic penetration is the BEST choice for a caster. This is your standard setup when playing any AP caster champ.
Seals: Health per level gives Karthus some survivability but is not nescessary needed. It may be replaced by other runes.
Glyphs: AP to have some starting power because it takes some time in this build to get started pumping AP.

Important things:
standard caster 9 in offence
4 points in max mana (max mana * 1.05). This will boost up your archangle's staff.
improved ghost and flash

First start with a mana cristal and 2 health pots. Then get tear of the goddess ASAP. Build archangels followed by the head of dead. You may want to get some survivability (health, armor, MR) after archangle's if someone is very powerfull. I then suggest building banshees or hunting guise

After the major AP items (you have 400+ AP). start bringing in game dependant items.

Playing karthus over 500 games I can say you will be ganked....A LOT! Get ghost and flash to secure your save way out. These spells are also great as finisher move combined with lay waste.

Farming tool = lay waste. (choose single targets)

chasing: defile on if in range, else OUT! You might need the mana

mid lane: always start mid(/solo). Try to farm and not harrass to much if the opponent can regen or out-harass you, if it is a squishy: LAY WASTE!

stacking tear of the goddess: Get golem if possible after getting tear of the goddess and use lay waste every 4 seconds, even if there are none minions or monsters or champions. Also in spawning pool etc. Make a last hit with lay waste and the mana cost will be returned by defiles passive and tear of the goddess passive. You will even get mana!

ulti: scan the map urself in solo play, let teammates call in teamplay. Tell them the exact damage of Karthus ulti--> lvl 1 -> <200 health (magic resist, 3sec health regen)

teamfights: try to catch as much enemys as possible with wall of pain to get max mana pen.

dead?: Never stop casting till the screen turnes gray!!

Not a lot of experience here, but Ranked is A LOT harder then normal game because U WILL get counter picked. Only pick karhus if you have last or second to last pick and switch solo/mid lane for better opponents.

REMEBER: Karhus can own many mid chars, but not the burst types that are unpredictable
(annie, katarina, etc)