KARTHUS:
With this guide made simple and easy. If you wish for more strategy or greater item variance, then please go to a different guide.

Righto! Well first off you'll want to buy a sapphire crystal or boots of speed. both are good choices. You'll want a tear of the goddess next, and depending on whether you want to move faster or hit harder sorcerer shoes or a blasting wand. Get archangels staff when you can after that, and then build for a rabadon's deathcap. After that, just spam archangels! Easy as that! Get a void staff if the other team is stacking magic resist but otherwise get 3 more archangels.
With this build you'll end up at over 1000 AP, and with magic penetration runes and sorcerer's shoes you should be able to dish out alot of damage.

Mejai's is also a viable option if you're doing well or feel you need that much more AP.

Magic penetration and ability power. Easy. Mana runes if you find you run out too easy, as the arch staffs turn mana into AP.

You'll want to max out lay waste as soon as possible, as it's your only real source of long range reliable damage. Get requiem when you can, and get at least one point in wall of pain as it's very good no matter what rank it is. Defile is second priority to lay waste, and is a solid way to get close range damage on enemies. Get 1 or 2 ranks in it before lay waste is maxed, then max it out.

I normally go with flash and clairvoyance, flash to escape nasty ganks or fed people, and clairvoyance to spy out requiem targets. Ignite is a viable option as well, and heal or clarity can be useful if you find you're needing health or mana in fights.

Lay waste:
Places a ball of explodiness at cursors location, blowing up after 0.5 seconds and dealing damage to nearby enemies.
The ultimate harass, you can use it early game to piss off the enemy laner/s or farm exceptionally well. Goes well with defile's passive. Late game will be a main source of damage along with defile.

Wall of Pain:
Slows enemy champions and reduces armor and magic res.
Use it to save yourself/teammates or to get that edge fights. Great for denying kills and also trapping escaping targets.

Defile:
Toggles on to deal damage to nearby enemies.
When using it make sure you keep an eye on your mana total, as it drains alot. Great source of damage in teamfights, make sure to keep it on when die for those 8 seconds.

Requiem:
Use it going into teamfights, or exiting teamfights. Very good at killstealing, but tell your teammates it was an accident if you do.

SO! Basically:
-Spam AP
-Cheese it when you see enemies
-AOE spam in teamfights
-DESTROY EVERYONE