This is simply my own custom Karthus build... focus power and destroy.

The quintessences, marks, and glyphs should be obvious. The magic-per-level seals are primarily for end-game Karthus, benefiting the three Archangel's in the build.

I went 21/0/9, focusing offensive. Follow the ability power related tree. Utility points compliment the build by giving bonus mana (again, useful for Archangel's) and granting some mana regen for mid-lane sustain. Furthermore, taking the Summoner's Insight allows you to use flash more often, which you can use for tactics, to pick up kills, etc. Finally, the point in Good Hands relieves a small bit of the burden of playing Karthus, who picks up those pesky runners after he "dies."

It is highly important to focus on getting the Tear first, to benefit from the bonus mana. Holding off on finishing the first Archangel's is actually beneficial, as Rabadon's and the Sorcerer's Treads do more for you (since you don't have the mana for AA to be useful quite yet). After you finish the first AA, you want to get some magic-pen in the mix. The Abyssal Scepter can be substituted with the Void Staff if the opposing team lacks substantial AP users. The final build caps out with 1086 AP.

Max Lay Waste as soon as possible. Take your ult whenever available... You need your second point to go into Defile or you'll quickly run out of mana in the beginning. Take the Wall of Pain as your third point to help you pick up a few kills, early on.

This will help you the most in picking up kills, as far as summoner spells go. 90% of players use this... don't be different, they use it for a reason. Possible alternatives include taking clarity instead of ignite to gain more sustainability (because those mana potions just don't do enough for their cost). Take ghost instead of flash if you really, really like ghost, but I recommend staying with flash--it's more useful.

Playing Karthus is relatively easy! Spam Q, benefit from your Archangel's staffs. Catch them with Wall of Pain, flash up to them, activate Defile and, again, spam Q. When at max build and, assuming that your opponent has no magic resist, your Lay Waste can do upwards of 1200 to a single target every hit.

Fun fact! With base MR being 30 on an opponent... 30 - 20(sorcerer's treads) - 20 (abyssal scepter) - 35 (wall of pain), gives your opponent a beautiful -45 MR. Meaning you do 20% more damage to them with magic.