Hi this is my first build so be patience please and leave comment below about what u think. That's my build for karthus and it works more then well for me. its simply OP.
Well early game u shold go mid. just hold back and spam ur Q as much as u can. his Q is an great range and really powerful skill combined with his W skill which reduces enemy's magic penetration. Your mana early shouldnt be problem because of his passive which gives him mana everytime he kills an unit (even minion). so if u run out of mana just go back to turret and last hit minions. Once you reach lvl 3 or 4 u can rush in for an easy kill. First harras him until he's 1/2 of his health ( which shouldnt be too hard if you're good with aiming your Q spell) then u just flesh in Use ur E then W and spam Q as much as u can and u can also use ignite if he somehow getaway. So he's 99% not able to get away but if he somehow got away u can just use your R for sure. that wont happen in early game probablly but still. U should keep checking your teammates if u can help them out with your R spell While it ain't on cooldown. Karthus is also very important in teamfights while teamfight is running just flesh into middle of fight and use ur E and W and spam Q. Even if u die. U still got ur 8 seconds to spam ur spells (cause of his passive). Also notice that karthus is really squishi champion. He's not good with 1-1. well you're probabbly gonna kill your enemy but you're probably gonna die too. so just be carefull. That's prolly all u need to know.

LAY WASTE - This is gonna be your main skill. It has low Cooldown so spam it as soon as it's available. It'll make tones of damage combined with your W spell as i said already.
Wall of Pain - Its a great skill which slows a target and it also reduces their magic pen if he passes throught. In team fights just throw it in middle of it to hit as many enemies as possible.
DEFILE - Great skill which does lots of damage. U can use it while enemy is trying to run from u . and u just follow him and it'll kill him cause of its range. u can also easily kill whole monion waves. but be careful this skill costs you alot of mana in early game.
REQUIEM - One of most fun spells in game in my options. Its great to use it while someone is trying to run away from u with low health and u just use it and it'll finish him off.
With the build at the end it'll make like 900-1100 damage.

Sorcerer's shoes - Great for its Magic penetration
Rabadon's deathcap - Great for its AP
Archangel's staff (x4) - It gives u an AMAZING AP at the end and it also gives u alot of mana.
At the end with this build u'll have like 1200 AP.
Enjoy it :)