SURVIVABILTY

If you have a problem with survivability with the first build you should buy a defensive item. Warmog's, Veil, and Omen are the best imo. Warmog's is the best overall when it comes to effectiveness per gold, but veil and omen are better in certain situations. (heavy caster, heavy phs respectively) I should have posted this earlier, but it is common sense to me. I sometimes forget that most of the people who seek out guides haven't been playing very long or just aren't willing to learn and would rather be led. There is nothing I can do to change this so there you go.

ITEM SYNERGY

I have come to the realization that most people can't understand the basics behind item synergy. Armor pen is only so good due to the order in which it is applied so limiting it to one or two items is ideal. If you get Last Whisper you should most likely not get another armor pen item. Bloodthirster is a needed item due to increased dmg, lane presence, and increased survivability. Infinity Edge is necessary since the increase to dps is immense. You have to build attack speed so that your AD is not useless and movement speed allows for better positioning. Crit is useful in triggering IE passive which in turn increases the amount of lifesteal you have. This is all common sense when it comes to LOL so you would have to be a real noob not to understand that IE should be on phs carries and that there is no better item for carries "overall"(it isn't for all carries just most). If you are feeding then start to build defensive items and I would suggest Banshee's Veil and Randuin's Omen. Both are very good for the gold involved. Generally don't build both, just the one that fits best for the current situation (most likely Veil).

The second build is pretty self explanatory. Thankfully I won't leave it at that since I am obviously wrong. Manamune is for mana and AD so that you can use your abilities to farm effectively throughout the game. Trinity Force is mainly for early game burst and movement speed. The other stats are a welcome addition though. Black Cleaver has good stats for corki since it now provides attack speed and gives a little extra armor pen more quickly. I have a veil in this build since this build lacks the force that the other has and needs to live longer in a fight. Phantom Dancer provides stats for the same reason in this build as the last. Hope that somehow helps now that you know how it works.

LATE INTRO
Hextech Shrapnel Shells - Corki's basic attacks deal 10% additional true damage to minions, monsters, and champions.

For an attack damage(AD) build this passive is too good. For all intensive purposes it used to function as a pure 10% dps increase, but now it is completely unmitigated damage. Corki is awesome.
Phosphorus Bomb - Corki fires a flash bomb at a target location, blinding targets in and around the blast.

Deals 80/130/180/230/280 (+0.5) damage to enemies in a target area. In addition, the blast reduces an enemy's chance to hit by 35% and reveals nearby stealthed units for 3 seconds.

Lets not forget about the 35% blind. While the damage is nice and the revealing of stealth units or brushed units is great, the blind allows you to go toe to toe with other heavy dpsers(Damage per Second) and come out on top.
Valkyrie - Corki surges to target location, dropping bombs that create a trail of destruction for opponents who remain in the fire.

Corki surges to target location dealing 60/90/120/150/180 (+0.4) damage per second to enemies in the fire left along his path.

Probably my favorite natural escape mechanic in the game. Not only is it quick and have a perfect range, but any champ who is chasing you will have to pay a price to take the most direct route. It also functions as a great dps ability against champs like Warwick who tend to forget what they are stepping in.
Gatling Gun - Corki's gatling gun fires multiple shots every half second dealing a quarter of his total damage, and shredding the armor of enemies who are under continuous fire.

Corki's gatling gun fires at all targets in front of him for 3/3.5/4/4.5/5 seconds, dealing a quarter of his total damage and shredding 1/2/3/4/5 armor with each shot. The armor reduction persists as long as the enemy is under continuous fire.

This is your game ending ability. It is so powerful that I feel absurd every time I play Corki. If you get a head start on the enemies and have a decent team to cover your back you can take pleasure in ending your enemies lives before they realize what happened.
Missile Barrage - Corki fires a missile towards his target location that will collide with the first enemy it hits, dealing damage to units around the target. Corki stores one missile every 10 seconds up to 7 missiles total. Every 4th missile will be a Big One.

Corki fires a missile towards his targeted location that will collide with the first enemy it hits. Each missile deals 125/200/275 (0.3 + 0.2) area magic damage.

This ability is now super AWESOME. It does loads of damage and is great for picking low health individuals off. It is also one of the best harassers in the game so push those lanes!

STRATEGY

Corki is one of my favorite Champs. AD is the only way to build him IMO. His (Gatling Gun) is so powerful and his other abilities have great synergy. Lets not forget that his passive is incredible with an AD build. One of the best parts about Corki is his natural escape mechanic (Valkyrie). Not only will it save you from Pantheon's Ult or save your *** if you are being chased, it is a really powerful attack unbuffed. (Phosphorus Bomb) is great for annoying champs who like to use the brush for cover and it blinds them by 35% while providing good dmg. It is safe to say when all is said and done that very few champs can out dps you. You still need to be wary of Nukes (Ryze, Annie,...), but even champs like Yi are powerless to your barrage. I generally use (Missile Barrage) for harassment purposes, but if all else fails, it is great to spam in a fight.

The alternative build is also very good, but I prefer to focus on (Gatling Gun) for power.

WHY NO AP

The main reason behind my dislike of AP Corki is because of the very poor scaling that you find with Corki's abilities. The highest scaling you find is with (Phosphorus Bomb) and that is a pitiful .5 or 1 dmg for every 2 AP. If those rates were higher I would play AP Corki all the time, but it just isn't worth it unless you want to feel inadequate. Thankfully no one uses AP corki now that he has gone from the least used champ in the game to one of the most popular.

So decided to update the Image with a new game with the second build. (seeing as it is my favorite) I basically never play Corki anymore. This is probably the first game I have played with Corki in 5 months. Probably not going to play him again anytime soon.

This buiild is very powerful. Sorry that it is just a normal game. Didn't think I would play well with Corki today.

OLD
Thought it might be good to post an example so I played a game today(06/29/10)
(Old Build Before BOTH Rune Nerfs)(Notice Date)

I know that it is bragging and all but this game didn't last till surrender time :)