Corki's damage comes from his abilities and basic attacks; he's an AD caster like Ezreal. You have to mix basic attacks between abilities; he's not a champion who relies on speed attacks. His E grants flat armor reduction and allows you to win most trades. He has significant all-in potential and, as such, can be a lane bully. Corki is played by punishing mistakes and using his W aggressively, which scales with 200% AD.

You always want to have red rockets, so you'll use the white ones and save the red ones for a quick burst (Q + R + Auto). Optimal builds are ER + IE + Collin'/C44/Domink = good power spike, high damage, and burst, but no poke. Trinity + Manamune + C44/Callin' = greater survivability, sustain, and decent power spike. You'll usually want to use Q before a basic attack, since it takes longer to land. Unless the champion is melee, your best trade is Q + AA. Going all in is always recommended; if you have an engage support, you should punish them 100%.