Generally, as corki adc you would need to rush essence reaver, because of the mana pool refund and cdr.

†rinity force - ad, as, cdr, health, mana, movement speed and spellblade passive. Every stat is good for corki.

beserkers greaves - adc boots
boots of swiftness - for attack move
boots of mobility - I generally wouldn't recommend this item, because your passive already gives you movement speed.

Statikk shiv/Runaan's hurricane - 30%crit and waveclear power

The bloodthirster - 80 ad, 20%lifesteal and shielding power

death dance - 80 ad, more lifesteal and, like shielding, reduces health loss as quickly.

infinity edge - 70 ad and 20% crit. Also benefits with passive.

Farsight alteration trinket - ranged people get range wards :) and, as you are squishy, ward places to stop ambushes or to know how many are around withought you having to be near. I have experienced warding a bush with warding totem trinket but they ambush me as soon as i ward, as they are close compared to a farsight ward.