Flash: Must have. Provides safety, repositioning and chasing. NEVER EVER EVER take something else.
Heal: Standard for ADC. You really need that safety and it gives you a really great movement speed buff for you and your support . Try to use this before flashing when you are getting ganked.

Trinity Force: Must have. 100% First item. NEVER start something else.

Sorcerer's Shoes: Since every of Corki's abilities are Magic Damage (except Gatling Gun) it is really good with magic pen. You can get berserker greaves (see situational items).

Infinity Edge: Take this as third item. The powerspike this gives with Trinity Force is so strong. Just imagine landing a Phosphorus Bomb + Sheen auto + the crit with extra dmg from IE. That is really strong.

Phantom Dancer For Atackspeed, Movementspeed and crit chance, Fourth item.

Bloodthirster: Get this as fifth item. It is really strong with the 80 AD and the 20% Life Steal + Shield is just massive.

Last Whisper: Without this you won't be killing that top laner with 200 armor or whatever it is. You really need this to kill juicy people.

Berserker's Greaves: If you are splitpushing alot and/or you need more single target damage, these are the boots you should buy. Otherwise just take Sorcerer's Shoes, but Berserker's Greaves are still an option.

Mercurial Scimitar: I fell in love with this, especially against Zed, Rammus, Warwick and Thresh. 80 ad is quite nice and the QSS active with a 50% movespeed buff is pretty neat.

Banshee's Veil: Generally the best choice, provides some health, antipoke and MR.

Guardian Angel: You are against a oneshot comb and your team can defend you after you respawn, the GA is the right choice for you.

R>Q>E>W
Points into Your ultimate whenever we can. There's nothing higher priority.

Phosphorus Bomb, higher priority because Corki trades best in bursts. You should be trading damage by Phosphorus Bomb. You want to trade damage if you know that you have the stronger matchup, or if their skills are on cooldown, or if they're going up for a creep.

Gatling Gun after Phosphorus Bomb. Don't max this first because it is unrealistic for us to get the full duration of your Gatling Gun off, unless they were afk.

Valkyrie last since it would be extremely suicidal to Valkyrie over people's heads for damage. The reason u take it at level 2 is just to be able to flee.

This is the standard ADC masteries that I use. You should always take 21/9/0 because 21 in offense is really needed for the extra offense (you dont say) and the defense is needed more than utility for surviving poke, teamfights and everything that involves the enemy team basically.

Why the mix armor and health?
You have very low base armor, so you don't want the enemy AD doing true damage to you due to you having next to 0 armor after they penetrate everything. You use health seals now because they give 8 per rune, which is extremely generous and makes them almost best in slot. I feel like this is a good mix since if we had too much armor, it has diminishing returns whereas if you didn't then they get closer to dealing true damage.

Why not lifesteal quints anymore?
With the changes to the runes and as well as the feast mastery, lifesteal quints have become less favorable and negligible to the point where you want to opt for full AD reds and quints instead. If you follow this guide you generally get a Vampiric Scepter-> Bloodthirster anyway, so lifesteal quints aren't that important anymore.

Due to the +(0.5 AD) scaling on Corki's Phosphorus Bomb, going AD in reds and quints is very favorable as they scale well with it.

Why attack speed?
With the buffs to attack speed quints, they're almost too good to pass up. You only get 2 quints of it because attack damage is still valuable to you.

:: Thresh::
My personal favorable supports at the moment due to the versatility of his kit. His kit allows for picking, escaping, and kiting. Usually if you had the option of a support, you can just ask for Thresh since he's almost never a wrong pick. During pre-6 laning, he generally does well against Leona. He falls behind Leona and Annie in terms of raw damage and kill potential at and after 6, but the rest of his kit more than makes up for this.

:: Leona::
Her kit is amazing for play making and fits most compositions. The only con to her is that if she engages, she's not likely going to be able to get out if the engage goes bad. During lane, Leona generally does well against Annie if she isn't afraid to engage. After lvl 6, she has stronger kill potential than Thresh you are willing to commit to that all-in with her.

:: Annie::
Her charged stun with tibbers is amazing for initiation and picks, which makes a prety good support. She generally does well against squishy compositions. During lane, Annie generally does well against Thresh.

Earlygame: Corki in early laning is pretty strong BUT VERY SLOW!!!. Just auto + Phosphorus Bomb and they can't go further. When you get sheen that will be even stronger. Just focus on winning on last hitting the farm.

Midgame: This is Corki's absolute strongest point. The mid game teamfights/sieging is where he excels. With Trinity Force + IE he is such a monster. Group with your team at this point.

Lategame: Corki's late game is solid, but there are ADCs that have stronger late game, but it is still really solid. You are still gonna be present in the late game.

Teamfights: Just hit things from the back and try to shred as much armor with your Gatling Gun as possible.

Conclusion: Corki is a really strong ADC and he is basically the new Lucian. Safe, mid game spike, a ton of damage and solid throughout all of the game.
Let's get into the pros and cons now shall we?

Pros:
Really safe
Lane bully
Good teamfight
Solid throughout all of the game
Provides A TON of magic damage, so he fits in a lot of team comps that need more magic damage.
Not alot of cons.

Cons:
Not the strongest late game