Introduction
Yah, this is no joke. Stacking Arch Angel Staff this way is way op. You end up with around 1200AP at the end... yea that's right... 1200AP. Your like "Oh, team fight? Oh, your team is full HP? BAWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW... Oh now your whole team has 1/4 hp and your carries just went BYE BYE.
Even IF they start stacking magic resist, or pick up 
Banshee's Veil to counter... Well that is one less slot for them... GG. Now their carries aren't doing as much damage OR they neglected MR all together and are going to die as soon as I cast my R... GG son.
Pros / Cons
--PROS--
You press R... You win.
Their carries pickup Banshee's Veil... You win (They do less dmg now).
They don't build Banshee's Veil, and instead build MR... You win. (^^^^)
You get to listen to them QQ about how OP Karthus is.
Mana? What mana? That blue bar does something? You Win.
--CONS--
A minion farts near you, you die.
You stub your toe on a rock... You die.
Super Squishy!
Runes
The runes are fairly self explanatory. The one thing I wanted to point out is that the first build uses level based ruins where as the second build uses set stat ruins. The reason for this is levels 1-6 on the first build, you should be passive. Since level 6 is the point at which the level based ruins exceed the set ruins, they should be taken instead. However, with build 2, is based around being semi-aggressive, and the early game stats allows Karthus to push the first kill and be annoying at the start so that you get fed for 
Mejai's Soulstealer.
Skill Sequence
Build 1)
This is a pretty typical Skill sequence for Karthus. Your 
Lay Waste is the first spell to max. Defile is taken next for the mana return. You want to get this early on for sustainability.
The difference from build 2 comes in at level 4. With build 1 you should take 
Wall of Pain to be able to escape an aggressive push. This is also needed since around levels 4-8 your AP will suffer due to building the 
Warmog's Armor. One point into Wall of Pain is all that is needed at low levels, after that the priority is:
Requiem > Lay Waste > 
Defile > Wall of Pain
Build 2)
You probably noticed that you don't take Wall of Pain until level 8. Two reasons for this:
1) You will have flash at lower levels to get last hits in
2) The idea is you are taking burst damage over CC. (Since you do more burst damage you don't need the wall near as much, since you will burn through them much faster.)
Since most of the fights will be done in the lane, one hitting on the fly with Lay Waste becomes problematic (Since creeps will move around). Although Defile is going to burn your mana fast at early levels, it also hits like a truck regardless of how many mobs you hit with it. Also, since Defile is now an extra rank higher than normal, it returns quite a bit more mana, and with your mana regen from last hitting creeps, you might not even need clarity before the other champion gets back.
Using Lay Waste to last hit a creep AND the opposing champion is a great way to get free damage in on the champ without burning through your mana. Doing this several times will give the impression that your hits are weak, giving them a false sense of confidence they might be a little more willing to stay in the lane longer before returning to base. They might even try to push you, thinking that you don't hit hard. This is the perfect time to stand your ground and start defile and start popping single target hits of Lay Waste, getting an early kill. You will notice they start fearing you and it's a lot easier to pester them in the lane.
Summary
Build 1:
For this build, you want to be passive for levels 1-6. Maybe getting a few jabs in on the opponent. However, you want to focus on last hitting mobs. With this build Last hitting mobs > Champion kills early game. You want to stay in the lane as much as possible and avoid early fights so that your level is not affected from down time being dead and running back. Getting the 
Tear of the Goddess as early as possible is key here. If you find yourself taking damage or losing mana, you might want to make a early trip back for 
Regrowth Pendant & 
Meki Pendant to increase your sustainability. However, if you have not taken much damage, or still have your health potions at this point I suggest going right for the tear before the Regrowth Pendant. Also I suggest always taking Regrowth Pendant & 
Ruby Crystal Over just taking the 
Giant's Belt if you can only afford one or the other on your trip back.
Even though you suffer from low AP while building the warmog's armor, you are still ultimately building AP through the Tear of the Goddess by converting it into 
Archangel's Staff. So the whole time you are building the warmog's you are also building up your mana pool from Tear of the Goddess, and converting all that mana into AP from Archangel's Staff afterwards. Once you get Archangel's Staff, you can start being aggressive since your survive-ability will be through the roof. At this point you should notice a seniffigent jump in the damage your doing. Then after Archangel's Staff you build your 
Rabadon's Deathcap and continue down the pattern stacking AP.
The low AP in the beginning is the reason you stay passive until at least mid game. As mentioned before, last hitting creeps and sustainability > champion kills for this build at early levels. Once your tear has had time to ramp up in late game and you get Rabadon's Deathcap you will melt faces off. Usually the opposing team will start building resistance to magic about this time. (Since they want to get there damage output first usually, they start building it late game.) This is where your last item comes in 
Void Staff which nullifies the little bit of magic resistance they have built up by now.
Build Guide
Discussion
Clarity
