corki has some very good burst dmg for a ranged dps and his gatling gun can take off a ridiculous amount of armor, essentially dealing true dmg or even extra dmg against most other carries. Corki also essentially has a built in flash with his valkyrie. His passive is also one of the better ones in the game as it gives him 10% extra dmg.
His disadvantage is that he is spell dependant as they make up quite a bit of his overall dmg output, if he runs out of mana, u lose

if u don get a kill with corki within lvl 3-5
ur doing something wrong
at lvl 3 u should have lvl 2 gatling gun and lvl 1 valk
valk in and turn on gatling gun
shoot them and if u think they r low enough, pop off ur summoners

i personally get exhaust and ignite, but this is because i play quite aggressively in the early game and try to get first blood, if you can get first blood at an early level, it gets very easy to zone the other champ out of exp range.
if you want to aim for survivability late game, it is better to take cleanse, ghost or flash.

In general Corki is pretty squishy and mana dependant, so i get:
3x HP quints
9x Mana regen per lvl blues
9x Mana regen per lvl yellows
9x Armor pen reds
This gives u a decent amount of health to start off and also lets u spam spells without needing to get a manamune, which lets ypu rush other items that give higher atk dmg.

i tend to pick up the doran shield first, it gives me enough survivability to last up until lvl 3 when i start to dive in and go for blood.
on ur first trip back, try to buy ur bf sword, if u cant, go for a doran's blade.
black cleaver is an absolute must as it gives even more armor reduction and it gives atk dmg, which is the most important stat for corki

corki, he can make the other team have -50% dmg reduction, what can be better
as mentioned b4, atk dmg is the most powerful stat u can get with him, his gatling gun gives dmg equal to half of ur total dmg per second