Not Updated For Current Season
This is my first guide. I decided to write one since I see so much bad Karthus play out there. If you have any specific suggestions, feel free to post them.
9 Offense- pretty self-explanatory, AP, CDR, Magic Pen
21 Utility- I don't get Spatial Accuracy despite using Teleport because... it's just not that good.
Awareness is needed to hit 6 and 11 earlier. Expanded Mind is 5% max mana... Assuming you have 2000 mana, that is 100 mana. Assuming you have 3000 mana, that's 150 mana. Not worth it for 4 mastery points.
Utility Mastery is great since you will rely on Blue buff a lot. Insight to gives your allies more mana in a sustained push, and Greed is just not worth it. 6 gold every minute, 180 gold over 30 minutes. Not very efficient.
Get Defile at level 2 so you can start getting mana back on your Lay Waste last hits. At this point, Lay waste does not do much damage and should be used for last hitting instead of harassing unless you are playing against a very dumb player. At 4, get Wall of Pain, and if you feel gutsy, you can try to land it on the enemy when they get close and Lay Waste them to death for first blood. Play defensive, but punish them if they think they can walk up to you without getting Lay Wasted a couple of times.
I get a 2nd rank of Wall of Pain at 12 or 13 so it is a little wider and better at catching people.
Requiem is great for intimidating your enemy team, slowing them, and finishing off opponents.
Defile is great for slowing enemies and dealing large amounts of AoE damage over time (almost 400 a tick fully farmed).
Wall of Pain reduces armor and magic resist and slows for 80% at rank 5. Self-explanatory. Great for chasing or saving teammates being chased.
Lay Waste deals double damage to single targets. Anyone not a tank (sometimes even tanks) will have a hard time taking a lot of these.
Item Build Order
*edited item build to illustrate build order*
Start with Amplifying Tome and Health potion. By the time you hit 6, you should have enough for Mejai's Soulstealer and boots of speed. Once you hit 6, it is CRITICAL that you constantly scan your teammates' lanes for possible kills/assists. Push your lane back, recall back to town, and get your Mejai's. Teleport back and continue to farm while watching the other lanes as well. At this stage in the game, unless the enemy is getting really aggressive, just play defensive and watch the lanes. At level 6, Requiem will generally kill anyone at 200 or below.
The next item you want to get is Rylai's Crystal Scepter. A lot of people don't build this item on Karthus. It gives him some more life, offers an AoE slow that helps chases or if your teammates are being chased, and is great later on if your Wall of Pain is down and you need to kite/chase someone down with Defile (you can toggle it on and off since the slow lasts for 1.5 seconds). A well timed Wall of Pain + Defile + Lay Waste spam will ensure no one can get away from you.
I finish my Boots of Swiftness after Rylai's. If you got the 4.5% movespeed quints and the 3% movespeed from Utility, Boots 1 is almost Boots 2 speed. With Boots 3 speed + quints/masteries and Defile that slows (thanks to Rylai's), you can outrun almost anyone chasing you, and chase down almost anyone trying to get away.
Zhonya next as in any team fight, people will want to kill you. It complements the Mejai's extremely well and gives your spells a bigger punch.
If the game is still going on, Guardian Angel is a good choice as deaths become insanely long (as well as giving some armor/mr and offering a buffer for your Mejai stacks). If the enemy is stacking Magic Resist, get a Void Staff. It really depends on how the game is going at that point.
Some people (actually, about 100% of Karthus players I've seen out there excluding myself) like to rush Archangel's Staff. While this item is good, it just doesn't fit very well into the build. Tear of the Goddess offers no AP until you actually finish the Archangel. Blue buff is an easy alternative for Defile spam. AAS also offers no survivability. The way I play Karthus, I am running up to people to get them in my Defile and Lay Waste the **** out of them.
Rylai's offers health and slowing ability, zhonya gives a huge AP boost and temp invincibility, mejai stacks are easily gained using requiem, and by the time you can get GA or Void Staff, AAS is probably not needed. If you don't know when to toggle Defile off, you're doing it wrong.
Play defensive in lane against strong mid opponents, keeping them away with Lay Waste pokes. At 4, attempt to kill them with Wall of Pain if you got them sufficiently low (60% or below). At 6, start hugging tower while last hitting and scanning the lanes for potential kills. Rush Mejai's to start getting stacks earlier. Do not be afraid to kill steal (unless they would easily die without your help-- then you just wasted your ult without getting an assist or kill). You are playing an AP carry with a snowball item, so kills are better for you.
At level 7 and up, you can solo Blue golem if you have full hp/mana (or clarity is up). This build relies heavily on Blue buff to spam Defile. Easily kill an entire creep wave quickly with Defile and Lay Waste spam. Always take Blue if it is up and you don't have it. Buy Elixirs of Brilliance to get more AP and CDR for Requiem.
If you're feeling gutsy, you can solo dragon if you're at full hp/mana with blue buff at 9. I've done it at 8 but I died the same time the dragon died. Probably doable at 8, risky though.
Always start with a Wall of Pain. The slow is ridiculous and lets you easily catch up, Lay Waste, turn on Defile, and Lay Waste them to death. With Rylai's, even if they avoid or get out of the wall, your Lay Waste and Defile will keep them in range and slowed for your team. Watch your mana carefully during teamfights, remember to clarity if Defile is sucking up your mana. Turn off Defile if you don't need it, or you won't have enough mana to Lay Waste. Activate Zhonya if you get targetted.
If Requiem is up and a team fight is imminent, as in, Amumu just bandage tossed into 5 enemies, and you are not close yet, Requiem first to deal its full potential damage, then jump in. By the time the fight is over, Requiem could come back up. If you're already in a fight, wait until you're dead or safe to Requiem. Any stun or interrupt (like your death) will interrupt the channel, using the cooldown without dealing the damage.
Questions/Comments are welcome.