This is my first build ever, where I'd like to show my way to play Corki. It's probably not the best, but I find it pretty fun. This will be as compact and short as possible. I'm not very talkative and I also think, that skill-set and item order is most informative anyway.

PROS
- nice damage output
- good farming

CONS
- pretty much mana hungry from start to finish if you play very aggressive (solved by golem buff)
- skillshots are not that hard to dodge for opponent

I prefer to use more defensive style quints, seals and glyphs to keep me in the line longer, while armor penetration marks greatly mashes with corki's abilities. You can however use any runes you want. If you prefer to do more damage you can go full AD or whatever :-) It's just a matter of opinion.

My masteries are logically focused on offense tree, where I use all available damage improving skills. The rest I put in defensive branch to help improve my durability a bit.

1. I start with Doran, I think it's the best choice to start even though it needs to be sold later.
2. I go for Berserker's greaves as second, improving attack speed and Corki's not that great movement speed.
3. Then I go either for Zeal or Sheen, depending on situation. The goal is obvious and it's also next step: get Trinity Force.
4. After Trinity, there's usually need for greater damage output, so it's time for BF Sword upgraded to Bloodthirster, giving good AD (needed for E skill) plus nice life steal.
5. As next step I usually go for Infinity Edge or Phantom Dancer. Exact order is depending mainly on who's in your opponent team.
6. I usually go for another Bloodthirster as last item, just to have insane life steal and even higher AD.