Aesdeyn here.

This is my first build, so let me know if there are any changes that need doing.

Karthus was one of the first champions I got to try, and I enjoyed him immensely...not for what I hear is the main reason, but rather because he is capable of dealing massive damage in a short amount of time - in almost any 1v1 he can take the enemy down before dying, if only barely.

For runes I usually choose the runes with the most Magic Penetration, Marks and Quints, with Seals of AP/level for a boost to mid-game damage and Glyphs of pure AP for early game domination.

However, this isn't the only way to play him. Where the 9 Mark of Insight are the best thing for Karthus to have, the 9 Seal of Force, 9 Glyph of Potency, and 3 Quintessence of Insight can be exchanged for a lot of different types to account for play style. The way I have it is more centralized on balancing early-mid game damage with no regard for survival, but getting more runes for pure AP gives a powerful early game to build earlier to lead to a strong late game; getting more runes of AP/level leaves the early game to being able to control the battlefield, but gives more power to late game fights.

I started with a basic 9/0/21 caster build, but took the three points out of move speed and put them into reduced death timers and max mana. I did this to be able to last in the lane longer early game and to be able to get back into the game faster once you're focused down, as Karthus is one of the first champions to be targeted.

To start, I either start with Boots of Speed or Amplifying Tome, depending on what the enemy team has capable of going mid. When they have characters like Ezreal or Cassiopeia, who have skill shot-type abilities that can bypass minions, I go with the boots to be able to avoid them. Otherwise I get the Amplifying Tome for early game damage.
Depending on how the first 6 levels go, the next item I buy is either Mejai's Soulstealer, if it's been a good start, or Tear of the Goddess, which Karthus needs to get eventually anyway to build his mana pool. If you can get to 995 gold before you get a kill, go back to base at the next possible time to start building Tear's mana.

By the time you reach the end of the laning phase, you should have an idea of what else you're going to want to buy. Void Staff, Rabadon's Deathcap, and Archangel's Staff are the items set in stone, but if you don't get Mejai's, that leaves 2 items slots to be filled by any of the following:

Rod of Ages: for the extra health and survivability, plus extra AP from both it and the mana for Archangel's.
Abyssal Scepter: in the case of a heavily AP enemy team, as well as extra damage from the extra magic penetration.
Moonflair Spellblade: this will help in a game with a heavily CC based enemy team.
and, my personal favorite,
Lich Bane: this item can turn a game around in the right situations, as with the AP you'll have built by this time it can get a tower down in 4-5 hits from full.