Korean Corki is NOT an ADC or an APC. He is both, which is why Corki is very strong if played correctly. Corki's abilities scale about half and half with AD and AP, meaning you should probably build a mix of both.
Phosphorous Bomb - 50% 50%
Valkyrie - 40% AP per second to a max damage of 100% AP
Gatling Gun - 40% AD per second to a max damage of 160% AD
Missile Barrage (a.k.a Rockets because they don't home like missiles) - 30% AP 20-40% AD
Big One - 45% AP 30-60% AD

First item to be discussed: Hextech Gunblade
How efficient is it? It has 45 AD 65 AP. Applied to the bomb, that is a 55 damage increase.
65 + 45 / 2 = 55
Each of the following, the infinity edge, bloodthirster, ond essence reaver, give a 40% increase. 80 + 0 / 2 = 40
Gunblade also has an active that slows enemies and scales off of AP. Gunblade is a very viable option for hybrid builders.

Choosing Essence Reaver over Gunblade. Gunblade has higher ability damage, and has a nice active, but people who choose to utilize Corki's innate passive ability of 10% bonus true damage may benefit from Essence Reaver's higher attack damage. Also, faster cooldowns mean faster rocket reload time, meaning more attack damage from trinity force. Choice is yours to make.


2nd item to be discussed: Bloodthirster
20% lifesteal and 80 attack damage - plus a shield from overhealing. This is very good on Corki because his playing style, described below requires closing distance and he may easily take more damage than other pure AD or AP carries.

3rd item to be discussed: Trinity Force
30 AD, 30 AP, 8% movement speed, health, enough mana to pull off 5 gatling guns or 10 rockets, and the passive spellblade.
Corki has extremely low speed, and the 8% movement speed helps a lot.
The spellblade's importance in the gameplay is explained below.
And everyone can use a little bit of health.


Information on Corki scaling.
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Note: Explanations are as of 8-23-2014.

Corki is best played as a Brawler. Corki's Gatling Gun and low range on Phosphorous Bomb (compared to Tristana, Twitch, or Caitlyn's super long range abilities), require that he should stay in low to medium range to maximize his DPS.

How do you maximize your DPS? Don't have a face to face shootout. Fire a few rockets, then fly over the target, making the opponent move to safety or get burnt, attack, gatling gun, attack, bomb, attack, then finish with rockets, making full use of Trinity Force. Not as easy as it seems because you can get ganked while your valkyrie is down. That's why you build Zhonya's Hourglass. You Zhonya then once your Valkyrie is available, you Valkyrie to safety, making enemies get burnt while following you, or leave you alone. Perfect for making counterattacks or getaways.

I chose Zhonya and Maw of Malmortius instead of the typical Guardian Angel because:
1. Zhonya has 120 ap = more damage.
2. You can break out of Zhonya any time you want.
3. Maw gives you a shield, similar to Banshee's Veil but has more damage, which increases as your health lowers, which means at maximum stacks, Corki can survive with his 30% lifesteal + attack damage.
In two items, you get armor and magic resist + 120 ap and 60-95 ad at the cost of HP. Better than Guardian and Banshee in my opinion.