Ok I hope everyone is ready for this, as I am about to describe in depth exactly how to dominate every game as Corki. The point of this guide is that a lot of people have trouble playing corki (and I did too for a long time), and he is often considered underpowered or too difficult to be a great champ. Well, I eat those people for breakfast, the way I play corki makes children QQ and hardened veterans scream OP. I'm putting this guide out there so that others can wreck face with this mighty mighty champion, to the point where he will need to be nerfed.

Q- Phosporus Bomb
GREAT skill, large aoe, great damage, blind chance, and utility all in one. It is your primary skill for harassing and can help you to farm, it also puts out good dps during early teamfights, and even does noticable chunks at the end of the game. If you are using it to detect stealth, pop behind you when your team is pushing if they have a twitch, put in in shaco's escape path, put it around low hp people if they have eve, and of course spray that **** all over akali's little shroud. It is just about the perfect size to make that skill useless for her. It can also be used to check bushes or dragon/baron if you need to, and can be used to check those places for wards (then kill them). Finally it lets you farm your heart out on teemo shrooms if you are ever lucky enough to be matched against one. When using this in lane, try to use it on the backrow creeps when the enemy champ gets close enough to be hit by it, or as a supplement to your auto attack for last hit timings.
tl;dr level this first, you need the damage

E - (yeah I'll come back to W) Gattling Gun
This is your late-game game winner. Every half second, you do half of your total attack damage to everyone in front of you and lower their armor by 5. By level 5 of the skill, you do this for 5 seconds (10 times, aka 5x your attack damage in AoE and 50 armor REDUCTION). This skill is how you kill people, and is great for taking jungle buffs. It is not, i repeat, IT IS NOT a farming/harassing skill early game. It's effect is really only compounded if you can hit multiple times and get the armor lowering to stack up... leveling and using it in lane is just far too expensive, that is what phosphorus is for. One very important thing about GG and corki is that corki CANNOT deal huge damage while kiting... that is what ashe is for. Corki is more like twitch in that he completely destroys anything in front of him, but does no damage whatsoever when running away. This means that positiioning yourself behind your team is very key, and the survivability plays a huge role in your ability to put out damage. This skill is also why we stack damage (because it gets no scaling besides your attack damage) and why BC/starks are so good... with GG+BC+starks you can be lowering an enemy armor by 130. And YES, that can take them negative (meaning you will start dealing +% damage on every attack or gattling gun hit). This is also the reason why no corki EVER should buy last whisper EVER EVER EVER. LW penetrates 40% of the armor that they have above 0. If they are below 0, it does nothing. If they have 10 armor, it is doing less than an armor pen rune... In general, unless the ENTIRE other team has more than 160, starks fervor will lower their armor by more than last whisper does. Let me explain using a metaphor... your terribad corki goes into a bar (enemy champ) and thinks that he's going to find some ladies (armor). This is a pretty standard bar so it has 80 single women (armor) Then, my corki comes into the bar and shows off his good looks (gattling gun) and tells some amazing jokes (blackcleaver). He does so amazingly that he leaves with 110 women, all 80 of the single ones and 30 that used to have boyfriends. So now your corki stands up on the table and starts yelling that he is going to penetrate 40% of the women in the bar, but the bar doesnt have any women, just 30 really pissed off dudes that drag him in the alley and beat the **** out of him just as my corki pulls away with his ferrari (starks fervor) that attracted 20 more girls while he was inside. /textblock.
tl;dr, don't gattling early, watch your positioning, use if for kills and jungling.

W - Valkarie

Hey its a great escape skill! Nothing much to be said about this, it isnt worth trying to use it for damage ever, just use it when you need to escape, or if you are ganking and KNOW you won't need it to get away, you can use it to close the gap and keep them in gattlinggun range. Early game, try to not have to use this ever because it is a huge amount of mana better spent on other things.

R - MISSLES YO

Very very nice skill, especially with the recent buff, you will use these in ways that are very similar to your phosphorus... for farming, harassing, and putting out some extra AOE damage in teamfights/ganks. ON FARMING WITH MISSLES: Farming with missles becomes a very important part of corki's life after level 6... one of the first tricks to learn is that a phosphorus plus one small missle is enough to kill the backrow creeps (and deal the damage to any enemy champ near there as well) When you have a BIG ONE ready, it does a lot of damage, so it can be worth saving it for when you know you will hit the enemy champ, either through normal harssing or while he is near his creeps. As a judge of AoE, the big one has about the same AoE as phosphorus. The other use for the big ones is that phosphorus + a big one missle will kill the entire wraith camp THROUGH THE WALL, meaning that after you madred's down the front row and phosph to kill the back ones, you can run over to the ENEMY wraith camp and kill it instantly, without leaving the lane. You can do the same thing to your wraith camp if you are somehow pushed back, but you should be a lot more free to leave the lane and take your camp normally (phosphorus + a couple auto attacks and/or a small missle)

With phosphorus, missles, and madred's procs, you will outfarm and outpush most mids, leaving you open to constantly grab the jungle creeps while still getting full solo exp. Golem is very important on corki because it lets you spam Q and E like mad, and lizard is equally important because it ensures that people cannot leave your gattling gun. As a general rule, golem is more important when laning or pushing, liz is more important when ganking or teamfighting. Grabbing a golem when you just have madred's/scepter or when you have wriggles is quite quick (fast enough you can get back to lane without losing any exp or tower damage). What I will often do is push the creeps to their tower, run off to take golem or lizard, blue pill to buy, then teleport behind to the ward in river behind them. (With wriggles, you are constantly able to keep a ward in the bushes by mid lane, so you are safe from ganks, it is also useful when pushing towers alone because you can set it up to warn you if anyone will be coming)