Karthus is probably best known for his ult which spans the map and tends to do some serious damage. He's more than just his ult though he's a game changing champ. A good Karthus levels up fast and helps push your team towards victory mid game. This build is designed to get you to level 18 well ahead of the other team, and allows you to be more aggressive in team fights instead of a nervous coward anytime a damage dealing champ heads your way.

Most Karthus players I see are basically jackals trying to mop up low hp champs, and running anytime a real predator comes their way. Yes Karthus has bad defenses, but that doesn't mean you play him defensive. You ensure he's involved in attacking the enemy as often as possible.

I'm starting off here cause this is probably one of the first things people notice about the build.
Yes I take revive. No I'm not a newb. It will allow you to play more aggressive not worrying about loosing XP from the occasional misjudgment, gank, or whatever get's you killed. Later in game you don't have to worry about dying in a team fight cause you can be right back up. There is a 540 second cooldown though so it's not like you're going to be feeding much w/ it.
Teleport let's you get back into battle asap, protect a tower, or just be anywhere you need to be. I basically recommend it due to it's versatility to work in several situations. Still if you have to change a summoner spell change this over revive.

Runes are pretty basic ability power, and magic penetration which help through the full game.

Masteries are focusing on the same thing, but adding some cool down, and magic penetration.

Skill Sequence is fairly basic.
Lay Waste is maxed out first. This will be your primary spam spell useful in so many ways once you get use to the timing.

Defile is next mostly used to farm, but is also pretty useful in close quarters combat. I try to use this only sparingly since it can really eat up mana.

Wall of Pain is only good for kiting. I only break it out if my team needs to run, or the enemy needs to run. Some people use it for initiating a fight which isn't bad.

Requiem (technically your first rank up priority) can be used in 2 different ways. Most of the time it's used to kill off low health runners. On the other hand I feel it works much better about 1/2 way into a team fight. Though you won't get as many kills you'll likely win the team fight. When using requium watch out for champs with Zhonyas Hourglass. It's a great shield against it.

While you won't really need the mana regen it's helps you build Archangel's staff which tends to be the first item major. I'm sure dorian's ring is tempting but you won't have health potions for the minor harass, and you'll either have to play less aggressive or go back early.

May not seem like a great item but it gives you decent ability power, and the mana to spam your Q more than you should.

Once you're out and about buy some shoes. I recommend Sorcerer's Shoes, but any pair seems to work.

Rod of Ages gives you a little bit extra of everything. Mostly it adds enough health for you to survive a little better.

Will of the Ancients will heal you up as as long as you have Defile up and a large number of creeps in the area. It also combines your ult into a half heal (assuming most the other team is alive).

Rabadon's Deathcap should be obvious for this build.

For your last item I leave 3 choices.
You're not being focused, and they have decent magic resist.
You're taking a lot of magic damage.
You're taking a lot of physical damage.

I don't think playing aggressive is the right way to describe how to play this build. Brave is probably more accurate. You don't need to piss yourself and hide anytime a champ breaks the front lines or stay out of effective range just cause your afraid some one will focus you.

You're gonna die sometimes accept it. You should be dying about 5 times a game. Less you're not being aggressive enough, and if more you're being too aggressive (or stupid). You should also be the first champ to level 18. Most my matches I'm at 18 when the other solo(s) are 15, and the rest are 13. Mostly this is because I'm gaining XP the entire time until I hit 18.

Your worst enemy is the gank. It's really easy to push a lane as Karthus, and you don't really have any skills to prevent getting ganked besides Wall of Pain. Getting focused in team fights is about as bad. Though if an enemy can get in long enough to kill you, and get out w/out dying that's probably more your teams fault than anything else.