Because you have nothing better to do ;)

But in all seriousness, many of the builds out there for karthus waste money and space on items that you don't need if you play right--honestly, who needs armor and tons of health as karthus? The range on his moves (not to mention his Ult!) is insane, and your enemies will scream at you and call you foul names in their frustration as you score Aces with your ult (at 675 AP, this happens). So, try it, comment on your success, and let me know how it works.
WARNING: This is for a mid-player only. Karthus on a side-lane is a whole 'nother animal, and unless he's taking it single (because of jungler or something), he won't be as effective. Also, I do not include runes because I'm not at lv30 yet, and Karthus so far hasn't needed any runes for what he does.

Pro's:
--High AP, high damage dealing
--Ult is insane with 675AP
--Enough mana so that active Defile doesn't run out when the enemy has 50 health left
--Uses stack item because you will EASILY farm all game

Con's:
--Still squishy-no armor! Can't engage directly
--Your enemies will rage when you kill them with ult
--Your teammates will rage when you 'steal their kills'

I only list one set of summoner spells because you don't really need any more :)
You will need this early game if you like to spam Lay Waste, which you should to keep your enemy at bay (and scared, or at least hurt). Last-hitting minions with this works, but wastes mana...and by level 13 you will be able to snap Defile on and off to superfarm minions, at which point you'll have a never-ending stream of money. This is one reason why this build is only for mid--you WILL out-farm your allies, and they will sometimes get mad because of it if you're laning with them. Clarity is rarely used mid to end game, but is a lifesaver in the beginning.

Teleport is used because it allows you to get back into the lane faster, and to save a tower if need be. Clean and simple.

Some recommend Flash but, to be honest, if you're getting close enough to where you need flash, you're not playing right anyway.

If you get used to the mana use and use it sparingly, you could also substitute Clarity with something else.

Ahh, the most fun part. I start the build with Sapphire Crystal and two Health Potion, which allow Karthus to be a serious pain in the rear when he's in-lane. With these and a careful eye on the opponent, you should be able to outlast him easily all through the game.

After your first kill, you should go back and buy boots or Sorcerer's Shoes and whatever else you can afford next in the build. This will give you the movement you need to dodge attacks (like, everything Brand does). Lane for awhile and kill the jungle creeps nearby.

Mejai's Soulstealer is the next item, and important at that. I'm debating switching the order that it's purchased in, but it's useful early and mid game. As Karthus, you will farm easily in the middle, so your stacks should go from zero to max in a couple minutes...and the cooldown reduction at that point is nice as well.

Rylai's Crystal Scepter is next. This is EXTREMELY useful for chasing down opponents trying to get away...whether you just smack them with Lay Waste, drop a Wall of Pain in their way, or trap them in Defile, your enemies will HATE you. If anyone complains that your only kills thus far have been the result of your Ult, this will silence them as you chase the enemy down.

Tear of the Goddess is just to add some mana in the meantime, not a big deal. It'll keep you in-lane longer and is essentially an item to grab anytime you feel low on mana.

Rabadon's Deathcap this is where the fun really starts. With this item, your AP jumps through the roof and you become a serious threat. This is where a smart enemy would grab Force of Nature or Banshee's Veil to try to fend you off. Then you get..

Void Staff which solves those problems. Well, not of Banshee's Veil but the other magic res issues.

Archangel's Staff This is the final, game-ending item. Ridiculous amounts of mana and AP will come from this, and it only gets better with mana stacks.

Playing Karthus is pretty easy!

First, a few things to keep in mind:
--NEVER engage the enemy first! Always let your team do it, then drop a Wall of Pain on the enemy and smack them for the kill.
--Use Wall of Pain along with Rylai's to have serious crowd control powers. Enemies trying to escape will be slowed to a crawl for a second or two, and that's usually enough time for you or your allies to sweep in for the kill
--Early game, use Lay Waste sparingly and don't use Defile at all. Defile's mana-cost-to-damage ratio at low levels isn't very good, and all you'll do is drain mana early on.
--Play defensively. If you get killed 2 or 3 times in mid without scoring as many kills, switch with someone else...nerfed karthus is almost useless except for KSing, and you don't want to piss off your opponents that bad.
--Blue buff will save you :)


EARLY GAME: This is an important part of the game. You have to use Karthus's amazing range on Lay Waste to ping the enemy and sometimes last-hit minions. You need the Sorcerer's Shoes as soon as possible unless your enemy is horrible. Just hang back, hit the closest row of enemy minions and don't be afraid to let your minions advance on their tower while you stay behind...use them as the meat shield that they are. Also make sure to stay out of the enemy's effective range (Jax and Xin can jump, Ryze can stun, etc.). As soon as you reach level 6, start checking the side lanes for enemies that you can end with your Ult. Early-game KSing is okay in this case, as its better for them to be dead than for your allies to overextend and feed.

MID GAME: This is where it heats up. When you get Defile to level 4 or 5, it becomes worth the mana drain to use it for farming and CERTAINLY worth it to hurt enemies. Punish those who rush in too close! This is the most important time of the game to farm, so that you can purchase all of those expensive items. Rabadon's Deathcap by this point is easy to achieve, and you should already have several kills from well-timed Requiem shots. Also don't be afraid to gank the side lanes--use Wall of Pain in the direction the enemy will go--thats BACK TO THEIR BASE--so that their only escape route is towards you or back at your allies.

LATE GAME: This is the end of it for the enemy. Full build by level 16 is easy, and certainly after a couple minutes at level 18. At this point you will easily dominate the game, unless the enemy has purchased all their items to counter you--in which the rest of your team will destroy in your place.

That's it--I'll add more to this section as I go, but please comment and let me know how it works for you!