No flash becuase of Valkyrie.
Valkyrie is just as easy escape as Flash in most cases.

-Heal?
-Why not, good if ganked.

Farm on botlane, only go BASE if you can buy an item or really need to fill up HP.

Need a sopport bot with good sustain (Sona,Soraka,Alistar or Taric is those who I like the most).
A support that have CC+sustain(Sona,Alistar and Taric) is even better for you as it increases the chances for kills and a safe way from the fight if needed.

When jungling start off at Red->Wraiths->DBL Golems->Gank if possible->Go base buy->Blue->Wolves, over and over again.
Gank mid or top first since they often/never buy wards in the beggining of a match.
A flash from the enemy team is a good gank from your side.
Try ganking the lanes without flash.

Replace Wriggle's Lantern to a BT(Blood Thirster) after you bought all the items.
Don't buy Tri-Force since it's overpriced and not relevant for an AD Carry like Corki.

Pros:
High DMG.
Long range with "R".
Valkyrie can save your life if you use it correctly.

Cons:
Easy to burst down(low HP).
Needs a good team that can hold the enemy team of on a distance while Corki shoots.
Hard to play on bot lane since there are many champions (i think Caitlyn,Vayne and sometimes Graves is good counterpicks to Corki since they have more DMG/Burst) that really can destroy Corki.