Hello, this is my first build here at this fancy website which contains so much inspiration regarding champion builds. Basically, I thought I'd contribute a little myself, and here you go!
This is also the build I used in the EU MobaFire tournament.

This Karthus build is mainly focusing on the gameplay for someone who goes mid or solo top, and is the way I've played Karthus ever since I started playing LoL, and is how I feel comfortable playing him. However, feel as free as you wish to disagree, but I'd appreciate it if you told me why in the comments. :)

Pros:
- Easy farm
- TONS of damage
- Very good contribution in teamfights
- A passive that allows you to mess around even if dead
- Annoying ultimate (especially early game)

Cons:
- Squishy
- Easily targetted
- No escaping skills

Alright, so I always start off with a Meki Pendant and two pots, because it gives me decent mana regen along with my masteries, as well as giving me the ability to heal up if I'm being harassed. The extra survivability that Doran's Ring usually provides, is partially covered by HP runes, as well as pots.

After that, I get Tear of The Goddess to get extra mana regen, which is needed, as you'll be using your Q for farm. Yes, sure, you've got a fantastic E-passive, but if you're going to harass as well, you might wanna get some early mana regen. Then you should rush boots, because, as you might've noticed, Karthus is really slow when it comes to movement speed, and with the extra magic penetration you get from the boots, you'll already be doing pretty nice damage.

Alright, so I've occassionally been getting mocked at for using an Archangel's Staff on Karthus, but what most people don't realize is that it provides mana regen plus mana stacking which really helps for late game teamfights, as well as giving you a decent amount of AP. I never run out of mana when I've got an Archangel's Staff. The thing about the mana stacking is that it get's so easily fully stacked, because of Karthus' Q-skill.

With only the AA, I only do decent damage, but what we really want is some AWESOME damage, right? That's why I usually rush a Rabadon's Deathcap after AA, to get the ability power needed to utilize Karthus' skills better.

So what? Now I'm a heavy damaging mage who dies in an instant. The thing is, after playing Karthus for a really long while (he's the first champion I ever bought), I noticed that there's no way I'm gonna survive a teamfight/gank unless I get to flash over that wall, as they usually focus me. In the beginning, I had really fun running around with 4 (yes, FOUR) Archangel's Staffs and 1k AP, but I died instantly and didn't really get to show of my powers, so I decided to build differently. However, I've never really been a fan of playing defensively, so the only full-defensive-purpose-item I've included in this build is Banshee's Veil, as it provides a nice passive, along with the MR and HP bonus. But it often happens that I do not buy Banshee's Veil and buy offensive items instead, like Abyssal Scepter, to compensate for the MR.

Anyways, the core is boots, AA and Deathcap, and as with any other build or champ, you need to pick the rest of the items according to your opponents. If they have 3 or more who have gotten themselves magic resistance, you get Void Staff, Zhonyas if they're doing massive CC output or they have a really strong AD carry, etc. However, I usually stay away from spell vamp, and I NEVER get Lich Bane because of the short cooldown on Lay Waste.

Kudos to all Karthus junglers (althought I never see you), but I'd rather rolfstomp mid. The important thing to remember when laning, and I'm sure you've heard it before, utilize your Lay Waste. It is the essence of laning Karthus, and will be the essence of your carrying. If you want to be good with Karthus, you need to be good at skill shotting your Lay Waste. A few tips: Try, as often as you can, to only single hit with it, as it's much easier to last hit creeps when you do it like that, because it hits the double amount of damage. This also goes for when killing or harassing other champions. There must be some Karthus videos up and running, and I'm sure if you study them you'll be finding interesting patterns on how to use Lay Waste.

As for harassing the opponent, you might find that some people easily figure out your pattern of using Lay Waste, therefore, you must be compatible to different types of patterns in which you use it, in order to harass successfully.

In teamfights, you might wanna position you in a way that is not outside of the teamfight, but not in the middle of it either. The reason for this is that you want to easily be able to hit people with your Lay Waste wherever they go, as well as get maximum "profit" out of your Defile. The more people inside your Defile, the more damage they take, and the more likely it is that you will win the fight. Also, remember to use Wall of Pain either right next to the closest and most likely escape route, or try hitting as many as possible with it, as it provides slow as well as a decent reduction in magic resistance and armour. However, to remain in this position in teamfights, you will need proper CC and a decent tank who won't let the whole enemy team target you.

If you start noticing the enemy team trying to target you, stay a little back and use Lay Waste and Wall of Pain into the center of the fight, and you'll still be able to do nice amounts of damage.

I prefer using Flash and Ignite as primary skills. Flash for obvious reasons, as I use it to get first blood or to escape, and Ignite to help me finish off people, as it reduces healing which makes it easier to kill champions who escape with your ultimate. However, there are some spells that would work rather decent when playing Karthus other than those two:

Ghost: I actually forgot to change my summoner spells once, when I discovered that Ghost actually works with Karthus, due to his freakishly low movement speed.

Teleport: Again, Karthus' low movement speed makes it harder to return to your lane in order to defend it or prevent loss of EXP, and Teleport might just be the solution to that.

Exhaust: I usually never get this, but against AD carries, this will definitely be convenient.

Thanks for taking your time reading this guide, and please drop off a comment below if there's anything you'd like regarding changes, disagreements or anything, really! :)