Best build path for Karthus?
Does anyone have suggestions? I want the most damage possible for the lowest amount of gold.
Does anyone have suggestions? I want the most damage possible for the lowest amount of gold.

For early items, you should consider your skill on the champion.
is the go-to if you're inexperienced on the champion, since a Karthus who can't land Q will have a weak lane.
is better once your mechanics are up to snuff, since a Karthus who can land every Q is a beast in the early game.
There are two schools of thought for main item builds.
Battle Karthus buys mana, tanks up, and wins teamfights by outlasting everyone on the enemy team.
This is where the old school Karthus build comes from.

It's a bit outdated, perhaps, but it gets the job done. Swap out
if you need a situational item like
.
If you're going this build, get a
on your first back and wait to skill your W until level 5, instead getting an extra point in E. If you're getting last hits to sustain your mana, you'll be able to use the increased E cost to sustain your HP indefinitely.
For this style, I reccomend running Aery.
Suicide Karthus is the exact opposite, building glass-cannon with virtually no mana or survivability, hoping to get people low in life before dying and finishing everybody off with a post-mortem Requiem.
I'm not exactly the most experienced with this method, but an example build would look something like this.

For this style, I reccomend running Predator; activate before pressing R, and the bonus damage will be sent to whichever poor soul is lowest on HP.
I personally go
with
. I play him top, i try to get a mix of dps and some HP to beef up.
is great ofc, i never built it cuz i dislike tear.