Phosphorus Bomb casts quickly and does good burst damage for a lot of the game. It can be used to harass an enemy last-hitting by autoattacking them and then quickly casting Phosphorus Bomb.
Valkyrie is a great farming, initiation, and escape skill. Use it wisely due to its long cooldown.
Valkyrie can be used to jump over terrain and walls in the jungle and with Flash, Corki can jump over 2 walls and escape most ganks.
Since Corki is a squishy AD carry, you should rarely use Valkyrie offensively. Use it to keep up range between yourself and the enemy.
Maximizing the damage and armor reduction of Gatling Gun will require Corki to be facing and attacking the target constantly for its four-second duration. It must be activated carefully to reach its full potential.
The armor reduction from Gatling Gun will be removed two seconds after the last tick of damage the enemy has taken.
Always watch your Missile Barrage quantity before entering a fight so you don't run out of it early. If possible, prepare a Big One. (Have at least 2 stacks towards one.)
Remember that Missile Barrage is an AoE skill, and it doesn't have to hit the enemy champion directly, it may be hit onto a nearby minion for the same damage. The Big One does more damage and has a larger AoE radius.
Because of the increased damage from his passive, getting a Doran's Blade at the beginning of the game allows you to deal a lot of damage. Starting with Boots of Speed and four Health Potions will give you more sustain and mobility, which is helpful when fighting strong poke and early gankers.
Getting a Manamune helps to deal with Corki's mana costs. However, it sacrifices late-game DPS.
Trinity Force is a key item for Corki because a Sheen proc can be frequently achieved using Missile Barrage and then apply it with auto attacks. This will strengthen mid to late game damage output (with a 200% boost to your base auto-attack) and provides some utility and survivability.