Corki was the yordle that got me from Plat IV to PLAT I, this guy can dish out massive amounts of not just AD damage, but also some AP damage. He is a great duelist, has (in my opinion) one of the best escapes of all ADC and he is just a really solid ADC. You can get close and burst somebody with your Q+E+R Combo, or you can stay in the back and kill them with your rockets. Corki is fun...

STARTER ITEM:
Dorans Blade and pot, the reason for this is the lifesteal combined with your passive is pretty sweet and will give you a edge in sustain.
LANING PHASE:
At the start just try to CS as much as you can, poke a little with your Q, if you have an aggresive support early, you can try to kill the enemys with a E+Q Combo.
When you hit lvl 6 you can pretty much clear a the caster minions with a Q+R Combo, and you can poke alot with your R and you can get some kills that will snowball you into lategame

TRINITY:
Trinity Force is a really good item is that it gives you some AP for your Q+R and some AD some movement speed and just gives you a nice start. Way to build it: Phage-Sheen-Zeal-Finish.
BOOTS:
You want to get magic pen boots because you will poke alot with your ult and your Q+R Combo will Melt the enemys with the magic damage.
BLOODTHIRSTER:
Get bloodthirster, it gives you so much lifesteal and alot of damage, + the shield is good if you duel their adc alot.
INFINITY EDGE:
Not alot of people get this on corki, but i feel like it is just so good. It might just fit into my playstyle, but again, this with you R+Q and you are devistating.
GUARDIANS ANGEL:
Guardians angel are just such a good defensive item on nearly all ADC, its just really good if you get caught you get a second chance, if their assasins nukes you, second chance. Its just a really safe item to pick up.
LAST ITEM:
For your last item you can get several different items, here are some suggestions:
Banshee´s veil, this if mostly for being against either a really fed AP ***ains, or a really annoying support such as blitzcrank, Morgana, leona, nami or thresh.
Last Whisper:
If your alot of AD champions on your team, they will build armor, get Last Whisper so you can still do alot of damage.
Lich bane:
If your full AD and you want some LULZ

Here is a little list of pros and cons about Corki:

Pros
Long Range
Mixed Damage
Long Escape
Super Burst
Really Fun
Wave Clear
No need to Facecheck

Cons:
Escape Cooldown
Bad Emote Spam
Team might hate Yordles