In this guide I will discuss the different possibilites you got with Corki, one of the most unique Carries in League of Legends and I will explain, why I took the Items/Masteries/Runes and basically my thoughts how to play this Champion.
However Corki was one of the most feared Carries out there, but was nerfed hard, so he is not that common pick anymore.

So lets go..

Hextech Shrapnel Shells is an awesome passive for AD-Carries, because truedamage cant be countered!

Phosphorus Bomb gives you some early poke and let you facecheck early on. However I feel like it is not that effective anymore since the nerf.

Valkyrie great in early game to play aggressive, but care in teamfights you should have this off cooldown to position new and dont fly in enemies there, because then you will be mostelikely dead. Just skill it one time on level 2 and then max it last.

Gatling Gun is maxed first, because it gives you greater damage output then Phosphorus Bomb.

Missile Barrage is great for harrasing in laning phase and works perfectly together with Sheen. Use it in teamfights as much as possible!

I decided to go for 21/9/0 Masteries instead of the more typical 21/0/9 Masteries for AD Carries. The reason behind this is, that i like to play aggressive in lane with Corki and want to farm safe at the same time.

So the extra armor and magicresistence I get there together with Hardiness and Resistance is really good for the early stage of the game.
Strength of Spirit helps you to stay longer in lane more then Perseverence .

The 21 Masterypoints in the offensivetree are chosen for obvious reason.. You are a Carry, who should do damage!

Greater Mark of Attack Damage helps your Hextech Shrapnel Shells so I take them over the more standart Greater Mark of Desolation, which are not bad either, but I think with Corkis amazing Hextech Shrapnel Shells these Runes fit better to Corki.

Greater Seal of Armor gives you the possibility to play more aggressive in your lane.
Greater Seal of Armor might be a good choice as well, but you will be much weaker early on and I feel like I dont need these Runes. You may give it a try, but I suggest you to go for armor.

Greater Glyph of Scaling Mana Regeneration will help you not running out of mana to fast.

Greater Quintessence of Attack Damage gives you some dominance in early and midgame. As the Greater Mark of Attack Damage, it stacks of course with Hextech Shrapnel Shells.

Early game

Mostlikely you will be on a dual lane (in best case of course with a supporter) were a Doran's Blade is the best option for early game. However, if you are on a sololane i would start with Boots and three Health Potion.
On your first way back you should be able to pick up a second Doran's Blade and Boots. For the rest of the money you might buy 1-2 Health Potion and 1 Sight Ward to be safe on your lane.
Then make your Boots in Berserker's Greaves for some attackspeed. If you face a lot of AP/stuns you can go Mercury's Treads as well.
Next will be a Sheen for the great damage combined with your Missile Barage.

Mid game

Finishing your Trinity Force will garanty you all Corki needs.
Right after this item I would go for Bloodthirster for great damage output and surviveability in fights. Some might say you should go here for Infinity Edge instead, but I feel it is to expensive to effort it there and it doesnt give you that much more damage for the higher price.
At this stage of the game you should have always an Elixir of Fortitude with you for teamfights.

Late game

Pick up a Banshee's Veil against Caster and Stuns (if you feel like you really need it, you might pick it up before the Bloodthirster as well).
Last Whisper will help you to shoot down enemy tanks much faster, while Madred's Bloodrazor will garanty you a lot of damage output in the later stage of the game.
Just make sure you get the buffs from the elixirs if you got some money to spend after buying a big item.

As a Carry farm is your bread and butter!

You should have like 100 cs per 15 minutes, so that you would got your itembuild completed after 45-50 minutes in the game, if the game lasts that long.

Laning phase

Try to get as much farm from the early on.
If you want to play aggressive, let the support engage then activate Valkyrie to the target get a Phosphorus Bomb on it and activate your Gatling Gun.
If you already hit level 6 shoot after every attack a Missile Barrage on the opponent.
Get Exhaust on him, if it is needed.
If you feel like it is getting to dangerous there just Flash out.

As you hit level 6 ,you should harras as much as possible with your Missile Barrage

Mid game

Try to keep your farm up! Be there where the farm is and just push as much as you can, but of course try to keep an eye on the minimap so you can see ganks or places where teamfights may happen next. Be there and help your team as much as you can!

Teamfights

Try to poke your enemies with Missile Barrage before the teamfight even starts so you got some advantage over them. Try to focus the most squishy targets first (in best case you get the AD-Carry of the team). If you fight 1v1 against someone, exhaust him and shoot after every attack a Missile Barrage to activate your Sheen of your Trinity Force.
If your position is not the best, get in a new position with Valkyrie and Flash if needed.
Of course you should have activated your Gatling Gun, if you focus someone!
You should always come from the back and dont engage with your Valkyrie, if you are not to sure about the situation. Of course you can hunt single low targets with it, but you need to be sure, that there is not a second opponent who stuns you or sth. like that!

Late game

Camp with your team, get the farm you need, get buffs, push the lanes etc.
Just dont do crazy 1v2 fights and be safe in your positioning on the map ;)

Corki is still a great Carry, who is a bit different from the others.
You got 3 different damageoutputs, so that noone is save from you.
If you managed to play a fine early laning you will be a great Carry for teamfights and lategame!

I hope you guys liked this guide and it helped you for what ever you watched out here :p

Please leave a comment below for good and bad things of the guide and your ideas :)