Hey there! My name is Leafbakko and I've been playing LoL since the Ahri patch, and most of the time I play AP Mid/Top. I play on the North American servers with my friends, my summoner name is Leafbakko. I really dont know what to say in guides because I don't want to make myself look like an idiot.

When farming, I try to farm with Lay Waste(Q) to avoid being harassed by the enemy AP late game, when I have alot of mana and there is no enemies in sight I farm with Defile(E) and Lay Waste(Q) together, this can almost instantly kill the minions. When harassing land your Wall of Pain(W) on the enemy for the slow and spam Lay Waste(Q) On them. If an enemy gets too close slow him/her with Wall of Pain(W) and if you are able to, get nearest as possible next to them and activate Defile(E) and spam Lay Waste(Q) This could be a easy kill. If the enemy gets away with low health be sure to ult!

Most of the time on items I try to focus on sustain when playing Karthus, for that I can stay in teamfights for a longer time and not die so often while still getting alot of AP.
Zhonyas Hourglass can save you at a deadly moment and gives armor, this does keep your Defile(E) on while used.
Seraphs Embrace takes 25% of your mana to make a shield, since you will probably be getting around 2.3k mana it will be useful in teamfights.
Will of the Ancients I find Spell Vamp useful on Karthus because of his insane damage with the MR reduction from Wall of Pain(W) combined with the spammable Lay Waste(Q).