For runes, I take Spell pen reds, Flat mana regen yellows, Flat MR blues, Flat AP quints. Why do I take these?
Spell pen: very standard and helps you do damage
Mana regen: helps you use your e early game a little more freely, and you can use wall a little more.
MR blues: Since your goal early game is to farm like a beast, flat MR blues will help you early game against strong harassers like Leblanc
Flat AP quints: Standard, helps you last hit
I typically run the standard 21-0-9 masteries that are shown above. Increased buff duration is OP since it lets you have blue for longer. But, against a strong lane opponent like Sion or LeBlanc, I take 21-9-0 with 3 in MR, one in armor, 4 in hp/level, and one in hp
As I mentioned earlier, there are two viable ways to play Karthus: Teleport/ Revive and Flash/ Exhaust.
If you're not taking these summoners, you're probably doing something wrong.
Flash: Standard on any champ, no real need to explain why I take flash.
Exhaust: Second summoner spell is the most disputed. I take exhaust in order to combo the slow with me wall so I can land a few extra Lay Waste and maybe get a kill early game. But most importantly, it helps me survive jungle ganks early game and helps me farm up. Remember to exhaust the person who is dealing the most damage, like Cassiopeia.
Why not Ignite?
Ignite is standard on almost any AP carry. But Karthus is special. In general, Karthus is played not as an aggressive laner (though he can be, I'll discuss this more under Laning). Karthus's goal early game is to farm as much as he can without dying, because late game he is an animal that destroys everyone.
Why not Teleport?
TP can be good, but as I said above, staying alive is most important early, and Exhaust helps with that.
Karthus does not lose to many champs in lane; he goes even with a lot.
But, he loses to: Ahri, LeBlanc, Galio, Talon, Anivia, Kassadin, Sion.
And even then, it's hard to really beat Karthus since he can farm from such a long range. I often first pick him and end up still winning the lane ._.
Your goal early game is to farm like crazy. I typically like to have around 100 cs by 14 minutes. You can last hit with your Lay Waste, since it refunds mana, and can turn on your Defile to quickly push a wave once you get blue buff. Once you get your Sorcs, you can start ganking other lanes with your wall and get a kill. Don't forget to farm enemy wraiths over the wall with your Lay Waste if your lane is pushed up ^.^
Oh yeah, ult when enemies are low, pretty standard.
A tower has fallen and people are starting to roam around the map. During this time, make sure you have ward coverage around the map and around objectives (baron, dragon, buffs); EVEN IF YOU HAVE TO BUY 5 WARDS YOURSELF BECAUSE YOUR SUPPORT IS AN IDIOT. Wards win games. True story.
Anyways, during this time, try to keep maximizing your farm. A good goal is ~160 CS by 20 minutes. Farm your jungle, and your opponents. If you finish your items, you basically win the game. Also, do NOT be afraid to initiate fights with a Wall of Pain-> Flash-> Zhonya's Hourglass combo. If you catch 3-4 people with a wall, you're probably going to win the fight.
Oh yeah, if you have a strong jungler and you have blue buff, go do a 3man baron with your top and jungler (after he gets an oracle, of course). Your E + Q shreds baron soooo fast. A 16 minute baron is such a huge advantage for your team.
KARTHUS WALL MAD OP RIOT PLEASE NERF. Your level 5 wall is ridiculous. The range and % slow on it is just... MAD OP. But anyways, flash on their AD carry, zhonyas, press q, die, q 2 more times, ult. Make sure to use your wall to initiate and disengage fights. If your wall is on cooldown, you're very vulnerable and your team should NOT engage. only engage with wall up.
Use this guide, enjoy free ELO. Be 14-2 in ranked like me. GG. Look me up, since this stat might be outdated (5/31/12)
Clannad is awesome.
Btw guys, watch Clannad. It's ****ing awesome. Just trust me on this one.