This is my first build, so bear with me.
I find this to be a very effective build for doing what Karthus does best: sweeping minions and reaping the souls of the weak.

This build relies heavily on mana-regen and the efficient use of Karthus' "E" skill. Thus, I start off with the Makai pendant, upgrading to Tear of the Goddess as soon as possible.

From there, spell vamp is VITAL for Karthus to stay alive and to take on any enemy champions that try to 1-on-1 him. Get that Hextech Revolver and you will be far less squishy.

Majai's Soulstealer is an option. If you can maintain at least 10 stacks, it's worth it's place on the list, though you may wish to swap it for something else (armor, perhaps) if it isn't getting it's money's worth. I like to get it early because it's relatively cheap, and the payoff can be spectacular if you have a good game.

You'll notice you're slow, and even though FLASH saves your butt repeatedly, you need to get some roller skates. Get the Sorc-boots.

From there, upgrade your Hextech to Will of the Ancients, and upgrade your Tear of the Goddess to Archangel's to get your mana and mana-regen up.

Next, save up lots of cash and get that Needlessly Large Rod, upgrading to Rabadan's.

You may now kill everyone/everything.

To further accomplish this, get the Abyssal Scepter (starting with the Negatron Cloak for defense) to deal even MORE damage to all those fools who forgot to get magic-resist items.

After this, I frequently end the round with lots of money to burn. Maybe I switch out Majai's Soulstealer for a Void Staff, or switch the boots for something faster. All depends on the needs of the game.

Karthus is a master at clearing minions. This is how he is most useful, this is how he makes all the money to get all those nice items.

Spam "Q". I almost never auto-attack unless I find myself out of mana. "Q" will be your primary attack for killing minions early game, and for killing Champions all the time. See, it hits double when they're standing all alone trying to hit you....

Once you have sufficient mana regen, you can use your "E" skill to clear minions my simply walking down the lane. Easy money. There could be 2, there could be 20, they'll all go down in a few seconds.

Karthus' ulti is an obvious choice for getting kill-steals. Whenever it's ready, scan the field to see if anyone's dying soon, or if your teammates are in the middle of a confrontation. Hit "R", and at the very least you'll get some assists, which will help boost the stacks from Majai's Soulstealer.

In more personal confrontations, hit them with "Q" like there's no tomorrow. ALWAYS make sure your "E" skill is on so if you die, you'll still be hitting them for your 7-seconds-of-post-death-mayhem. I can usually kill off anyone who kills me. Keeps it fair.

His "W" skill is also vital when dealing with Champions. Use to slow them as you run away from them. Or use it to slow them when they're running away from you!
Even better is when you use it suddenly in a team-battle, and suddenly all the enemy champs are like fish in a barrel, stumbling in slow motion as your team wipes them out.

FLASH is primarily for escaping, as Karthus tends to be on the slow side when retreating (everyone else has boots AND Phantom Dancer, it seems)

Also useful to launch into battle unexpectedly when your team needs your help.

TELEPORT because it's just so darn usefull to get to those hordes of minions that others aren't good at clearing quickly. While it takes some champs forever to clear a crowd of 20 minions, it takes Karthus about 4 seconds, and suddenly that lane loses its momentum

Also integral to this build is the Ancient Golem buff! Once you get the Hextech revolver, he is easy enough to take down. Then you will have the mana-regen needed to do serious farming, as well as some much-needed cooldown-redux so your Ulti will come more often.
Try to have this buff as often as possible.