When you complete the following areas in either Adventure mode or Championship mode you will unlock the indicated skin. These skins will then be available for selection whenever you start a new game thereafter. You begin the game with the Original and Vintage skin.

Unlock Ancient:
Clear Area 1

Unlock Hardcore:
Clear Area 2

Unlock Super Hardcore.
Clear Area 3

Here are ALL the achievements and what needs to be done to get them.

Area 1 Clear (5)
Cleared Area 1

Area 2 Clear (5)
Cleared Area 2

Zero Deployments (5)
Area cleared without deploying any satellites

Adventure Area 2 Clear (10)
Cleared Area 2 in Adventure mode

Area 3 Clear (10)
Cleared Area 3

Area 4 Clear (10)
Cleared Area 4

Area 5 Clear (15)
Cleared Area 5

One Life Only (15)
Area cleared without losing a single fighter

Adventure Area 3 Clear (20)
Cleared Area 3 in Adventure mode

All Championship Areas Clear (25)
Cleared ALL areas in Championship mode

Adventure Area 4 Clear (30)
Cleared Area 4 in Adventure mode

All Adventure Areas Clear (50)
Cleared ALL areas in Adventure mode