Welcome to my build ladies and gentlemen. Karthus is my main character and definitely one of the best team fighting champions out there, if played correctly. Martyr Karthus is exactly what it sounds like. You will die. If you want your stats to look pretty, go find another guide, or preferably play a different caster.
Now, that being said, you will get a lot of a kills. A lot more kills than deaths. You will be picking up double kills, triple kills, often quadra kills, and every now and then penta kills, every time you die. Especially double kills. You'll be dropping double kills like it's your job.
So let us begin.

Magic Penetration is an extremely important stat for karthus. You will be making yourself plenty known, and any half competent team will build magic resistance and build it early to counter you. Magic pen blues and reds will suffice, especially along with the sorcerer's shoes.
AP yellows will give you an excellent boost early game. You can also replace these with health seals, as karthus is a particularly squishy champion. The boost of health early game is extremely useful, and there is often enough ability power from the AP quints to hold you over until you get your blasting wand.
Your quints should be two AP quints and one revival quint. The revival quint, plus good hands, plus your passive, death defied, will make your respawn timer extremely short. This is nice, since you are going to be dying.

Masteries are self explanatory. 9 in offense, 21 in utility, always.

Here we go. Start with a sapphire crystal and two health potions. Put your first stat into lay waste Lay Waste.

Karthus should always take mid.

Start off playing conservatively. Remember lay waste =/= mana waste. Use it to finish minions off, and use its AOE to harass a bit, but not too much. If you can hit a champion and finish off a minion at the same time, do it. Once you level defile, you will regain a small amount of mana every time you kill a unit with it. Do not turn defile on to farm. It drains mana too fast and does not do nearly enough damage at this point. at level three, level your wall of pain Wall of Pain.

Once you hit level four, the real pain starts.
Use lay waste to harass. Remember, if only one unit is in range of lay waste, it does double damage to that unit. Try to catch the enemy champion alone with lay waste, it will do huge amounts of damage at level four. Continue to harass and farm until level six.

Once you get requiem Requiem, let your teammates know. Now kills also become easy as pie in mid lane. Drop your wall on an enemy champion to start. The slow from the wall will make it easy to land your lay waste, and the reduced magic resistance will make it hurt even more. Ignite them, and lay waste until dead, with some help from defile. If they escape, finish them off with requiem.
Easy peezy japanesey.

On your first trip back to base, pick up your tear of the goddess, your blasting wand, and boots of speed. If you can't afford both the boots and the wand, just get the blasting wand, and if you can't afford that, just get the boots. But your tear of the goddess is always item number one.

REMEMBER TO FARM. You should easily have 100 minion kills by 20 minutes.

This is where Martyr Karthus will really shine.

Karthus is really a bit of a cheap character when played right. He has fiddlesticks' ult on a toggle, and he can cast volatile spells after he dies, not to mention his ult that can kill you anywhere on the map. So exploit it.
Open up with a wall of pain. At level 5, it is an 80 percent slow and reduces a healthy amount of enemies' magic resist and armor. Try to cast from afar, but if the fight comes to you, welcome it with open arms. Turn on defile and shred their squishies. Do not run from the middle of the fight, as long as your team is still there. You will get focused, but you want to aim to die in the center of the teamfight. Now, cast lay waste with your defile on. Don't forget to ult. Remember, you get 8 seconds to cast while you die, and your requiem is a three second cast. You can do the math, just make sure you get it off.

Got a tank that doesn't want to initiate? Turn your AOE on and flash into the center of their team. Your 80% slow will make sure your team follows you right in and their team can't do anything about it. You'll die. You'll get a bunch of kills and assists. Alternatively, you can use requiem to initiate team fights. Your team will most likely be way more willing to fight if everyone on their team all of a sudden has 800 less health.

Karthus is not a good pusher, but he can take towers down FAST with lich bane Lich Bane and a ton of AP.

Allied Team
Specific allies are not often important, as long as you have enough physical damage to make sure the enemies cannot just stack magic resist.
Also, be sure to have a good pushing team. Karthus might be able to drop two, three, or four enemies in a teamfight, but if nobody on your team can push to take advantage of it, You will be dying for nothing, and it will defeat the purpose of your play style and your deaths. I don't know how many times i've played with a team that would not push without me, their carry, present. Those are games you will lose every time.

Enemy Teams
Karthus is a great champ, and will get kills on any team he plays against. However, he can be easily countered by certain champions and items.

Here are some champions to watch out for:
Shen is a very strong tank. His Feint can ensure he is safe from your ult and Stand United can ensure that a teammate also will be safe.
Vlad. I hate this champion for so many reasons. But in the case of karthus, he can simply Sanguine Pool out of your ult. make sure to watch for him to use it. If he's deep in ****, he'll pool out. as soon as he comes out, drop a train on his head.
Kayle can heal a target, and make it so they take are invulnerable for up to three seconds. You can see the obvious problems.
Leblanc is very difficult to lane against. She can out put huge amounts of damage and move away from your wall of pain slow/lay waste follow up.
I don't know who decided that making a character who has superior range to everyone and does a ton of damage at any point in the game without items was a good idea, but it makes for a very difficult lane. Lux can harass a lot better than karthus can, and karthus can't take that much abuse.

Here are some items that can cause you a lot of problems:

Remember, use your ult to pick off enemy champions whenever possible. If a champion is low enough to kill them, kill them. don't save it for something else.

Do not ult if you're close to dead and not safe. Dying will interrupt your ult.

Do not ult if you can be stunned, taunted, or silenced. There is an extremely long cooldown and wasting it is the worst thing you can do.