Hello and welcome to my way of building karthus throughout a normal 5v5 game. Be overwhelmed and try it out yourself, and leave your comments and thoughts down below, so I can improve my build on any points you feel something else would fit better. Note that my item sequence is based on playing on mid, not on bottom or top.
Enough chatter lets go to the build.

Pros

Nice Ultimate, the best in the game for me.
Nice damage late game.
Awesome harrasser with lay waste.
Very good slow (80% at skill level 5)

Cons

Very squishy early game, you have to watch out if the enemy team has a jungler.
Needs to have a good early to be nice mid/late game.
People will flame your ultimate.. :P

Farming with karthus is the main thing you should focus on in the laning phase, when the minionwaves encounter, stay behind your melee minions and focus on the 3 caster minions with Lay Waste. Harrass your opponent when he/she comes too close, and be sure to only hit them to get double damage from Lay Waste. Be sure to last hit them with your lay waste as much as possible because if you do it well your enemy will be outfarmed easily, especially when you get Archangel's Staff, Defile is an option too then, but only toggle it on/off when it does some decent damage, otherwise it'll drain your pool to 0 for 6 minions, and you don't want that, obviously. Farming with karthus is, as I said, a really important factor to become good in mid/late game.

I chose Flash and Teleport mainly for some survivability and to teleport to farming spots in mid/late game, you know, these bunches of minions pushing a lane. I also find it useful to get quickly back after you had to go back, or after you died. A nice teleport + a Wall of Pain and some Defile/Lay Waste = kill in Teamfights, if not blow your ulti afterwards, that should almost always secure a kill for you, or for a teammember. You could also choose to get Flash and Ghost, but I think Teleport fits better for Karthus, but that's just my vision on how I play him.

For runes I take Magic Penetration marks, Mana seals, ability power glyphs and magic penetration Quintessences, which aims for penetrating the magic resist they stack, cause people always start doing that when they see karthus' ultimate doing fine. I also build a Void Staff later on to add another 40% magic penetration, this helps killing tanks with your ultimate. Nothing is more annoying than a tank getting away on 10 hp after you did your ultimate, the extra magic pen from runes/items covers this problem.

My masteries in utility are aimed on laning durability, grabbing regeneration, teleport, flash and some cooldownreduction on all abilities. In Offense I go for some more Ability power and 15% extra magic penetration, which is amazing for Karthus. Furthermore I want to add that I use this masteries for almost all mid-champions with ability power I play, so it's kinda general for me, not only for karthus. Feel free to adjust some masteries to your own style, you should really add your own touch to this build to fit your way of playing karthus, I just give you advice on how you could best build him.

I start with a Sapphire crystal and 2 health potions to cover my durability in the middle lane. Health potions to be able to stay longer; Sapphire Crystal to be able to harrass some more. More mana = more Lay Wastes = More Farm = More money, win-win situation if you ask me.
I pause building archangel's for Sorcerer's Shoes, to give me some early magic pen and movement speed. After the boots I continue with Tear of the goddess for some extra mana and later on I upgrade it to archangel's staff for more AP and mana. From here on you should decide what to buy first; depending on how good you can farm. If it goes well, you take rabadon's deathcap first, if it doesn't go well you grab the rod of ages first, just depending on the enemy team and the progress of the game, you should decide this yourself.

The skill sequence I selected is based to get most out of your abilities, maxing Lay Waste at first for some nice damage to harrass your opponent on mid. Don't fear going out of mana, because if you kill a unit with your basic attack or any ability Defile's passive (only works when defile is toggled off) grants you some mana back, so maxed Lay Waste returns you a lot of mana when you are good at last-hitting. Then I aim for defile at second to cover farming possibilities after the laning phase, all aimed to get gold as fast as possible for your itembuild. Then I continue with the Wall of Pain for a 80% slow, chasing people who are slowed for 80% with defile + lay waste is epic win. Obviously grab a point on Requiem at lvl 6-11-16 as usual.

You should never engage in teamfights, you will be focused almost every fight because your damage, ultimate and your slow is the first thing enemies want to take out, improving their chances to win the teamfight. Always stay back and put lay wastes under your enemies to damage them and if the yescape, watch their HP and decide whether to use your ultimate at the end of the teamfight or not, always look the enemy's HP and your ultimate's damage and calculate a difference of 10% for resistances, if they are below your ult damage you should Kill them, if not then don't use your ultimate, obviously. Coordinated ganks from bottom and top always depend on your slow, if you place it behind your enemy he has no way to escape because of the slow and this should almost always ensure a kill.

On the below 2 screenshots you should watch my Minionkills and my kills on champions, always tell your teammates to write anything you wish them to say if they need your ultimate for a kill, something like Req! could do (Req from "Requiem"); it should be short and descriptive so you wont lose time, the enemy could have recalled already if it takes too long, also keep an eye on the bottom en top lane yourself to secure kills on enemies who would otherwise get away. The screenshots :