This is my first build on Corki, I'll try to update and improve it often.
I've got to say I'm not a PRO with any of the Champions but I can sometimes make the tide turn using Corki. Play him well, that's all I ask.
This build is all about the efficiency of Corki and his skills.
I don't like the auto attack builds or the full AP based (which are the worst) so I've experimented on him to eventually achieve this hybrid AP/DPS Build.

AoE = Area of Effect
MP = Mana
HP = Health
SV = Spell Vamp
LS = Lifesteal
AP = Ability Points
DPS = Damage per Second
AD = Attack Damage
MRes= Magic Resistance
Arm = Armor

PROS

Good ganker and harraser against squishy champions
Great Farmer
Good DPS as AP champion
Spammable skills (Eats your mana though)
Good escape method with damaging trail
His quotes are Codes from WWI (Lots of insults^^)
Nice team battler

CONS

Squishy
Slow (Since patch) 310>300
Item Dependent

I'm building 9-0-21 because you'll want your skills to be spammable and efficient.
I've got nothing more to tell about this it's your choice not mine but I think it's best like many others do.

For Runes I choose:
Armor Penetration Greater Mark of Desolation - Armor Penetration Greater Quintessence of Desolation - Magic Resistance - Armor
Since Corki is squishy I wanted more MagRes and Armor plus some Armor Penetration to add a destructive effect in combination with the Black Cleaver

Early Game

When you're ready go for the items described in the Items paragraph.
I suggest taking a single lane but if you've got a better mid lane champion, let him go mid.
Once the game really started, try to get last hits on minions for farming. If the other Champions engage you or your ally use .
When the "Bogey" makes a mistake by engaging at close range, use to shred their armor, following with some Auto-Attacks to finish them off.

Mid Game

Entering Mid Game makes you a bit more vulnerable. Getting the
can provide you some better survivability. Use the , and combo will bring most champions, except tanks, to the half of their HP. Since you are very vulnerable, the outcome of your score is not decided mid game. Keep in mind to spare ready if you have to get out.

Late Game

Late game is the deciding phase of a match for Corki. With great damage output for skills as Auto-Attacks you will be able to win almost every teamfight late game.
Spraying the battle with and will deal huge damage. Single battles (1v1) will be tougher but still winnable. Use the and combo, Auto-Attack for the remaining HP and (this is the part for skillshot advanced players) finish him off with if he managed to get away.
BUT don't ever,EVER use to chase unless you're fully sure that you're safe. If you don't managed to get your kill, you should train on the skillshots.

Early Game

At the start of the Game you should go for
EITHER , OR
Pick Doran's Blade for some more survivability,
Boots of Speed if you want some early maneuverability
or Long Sword for a great early damage output due to his passive. Both the boots and the long sword have a further build in this guide.
Later on get or depending on the enemy team build.

Mid Game

When you've farmed a bit you should buy:
, and
Manamune is a great item to prevent a refueling round (Recall) on the wrong moments and it gives some AD on top.
The Hextech Gunblade provides AP, AD, SV and LS ; Nice item for Corki I'd say. It gives his skills a boost, increases his AD and gives some survivability. What's not to like!
(Ok,ok it's expensive and yadayadayada but you've got gold for something)
The Black Cleaver for some Attack Speed and Damage. In combination with it fully shreds the armor and makes you the Ultimate tank destroyer.

Late Game

After some good killing and farming you deserve some more so sell your Doran's Blade.
Get
and /
Trinity Force for some overall power and survivability.
Banshee's Veil/Force of Nature for some extra MRes depending on how strong the enemy AP Champions are. If they're not take Force of Nature for extra maneuverability.