Corki is probably one of my favourite champions to play. He has awesome abilities to nuke down almost every opponent, Plus he has an awesome escape ability.

Early game is by far the hardest for Corki, this is why i go mid. Early game is all about last hitting minions and harassing the enemy "midder". i usually have around 150-200 minion kills at the 20 minute mark. At level 6 when you have your ultimate you can go kill the wraiths when your lane is pushed to enemy turret.

Here is a picture of some of my games, and the stats i've gotten.

I start of with a cloth armor, 2 health potions and 2 mana potions.x2x2

And as soon as i have 700 gold i recall back to buy Madreds Razor'sthen i use teleport to get quickly back into lane

As the next item i buy Wriggles Lantern
this is probably the toughest time for you, because after you get to level 6 and get some attack speed from items you will totally own mid lane against almost every champion.

I start of buying a cloth armor and 2 HP and MP pots, this is great for taking the mid lane.
When i got enough money i recall back and buy wigglers lantern for easy farm on minions. i then buy berserker Grieves for the extra attack speed and movement bonus. The Black Cleaver is awesome for tearing up the armor the enemy has. And Infinity edge and Bloodthirster is just for the bonus dmg and lifesteal that will make your Corki so op that the enemy team will cry!

If a game takes so long that i get full items, i sell the Wriggles lantern and buy a second Bloodthrister or a trinity force.

Pros:

Great damage out put late game.
Good farming abilities.
Valkyrie is a great ability for escaping.
But mostly he's fun and epic :D

Cons:

Very squishy early game
Requires alot of farming early game.
Goes easily down if stunned.

Hextech Shrapnel Shells

(Innate): Corki's basic attacks deal 10% additional true damage to minions, monsters, and champions.

Phosphorus Bomb

(Active): Deals magic damage to enemies in a target area. In addition, the blast reduces an enemy's chance to hit by 35%.
Cooldown: 8 seconds.
Range: 600.
Area of effect: 300 (estimated).
Cost: 80 / 90 / 100 / 110 / 120 mana.
Magic Damage: 80 / 130 / 180 / 230 / 280 (+0.5 per ability power).

Valkyrie

(Active): Corki surges to target location, dropping bombs that create a trail of fiery destruction for 5 seconds. Opponents take damage each second they stay in the trail.
Cost: 100 mana.
Range: 800.
Speed: 650.
Cooldown: 20 / 18 / 16 / 14 / 12 seconds.
Magic Damage: 60 / 90 / 120 / 150 / 180 (+0.4 per ability power) per second.

Gatling Gun

(Active): Corki's gatling gun fires at enemies in front of him every half second for a short period of time, dealing a quarter of his attack damage and shredding the armor of enemies who are under continuous fire. Armor reduction stacks and lasts 2 seconds.
Cooldown: 16 seconds.
Range: 600.
Cost: 60 / 70 / 80 / 90 / 100 mana.
Duration: 3 / 3.5 / 4 / 4.5 / 5 seconds.
Armor Reduction: 1 / 2 / 3 / 4 / 5 .

Missile Barrage

(Active): Corki fires a missile towards his target location that will collide with the first enemy it hits, also dealing damage to units around the target.
Corki stores one missile every 10 seconds up to 7 missiles total, the time to respawn a missile is affected by cooldown reduction and does not progress while having maximum missiles. Every 4th missile will be a Big One, dealing 60% more damage with 100% splash damage. After dying, Corki will respawn with 4 missiles.
Cooldown: 1.2 seconds.
Range: 1,200.
Area of Effect: Regular (150); Big ones (300)
Cost: 30 / 35 / 40 mana.
Magic Damage: 120 / 190 / 260 (+0.3 per ability power) (+0.2 per attack damage).

I take Teleport and ghost, Teleport for getting fast back in lane, not giving the enemy more exp. and ghost for situations where you need a quick escape and valkyrie isn't just enough, or for when chasing.

Like i said in the in the introduction, farming is really important for Corki. I've been killed in mid an still over leveling just by getting alot of minion kills!

Madreds Razor's is a great item to start of with just because it improves your farming.