When you complete the indicated tasks to unlock the corresponding alternate costume it will be available for selection when you start a new game.

Unlock Spud Of War costume:
Complete the game on any difficulty. This costume increases your Magic skill.

Unlock Mime Of War costume:
Complete the game on God mode. This costume gives 10x Red Orbs and makes weapons invisible.

Unlock McKratos Costume:
Complete ALL 5 missions of the Challenge Of Hades.

The following features in the game become unlocked when the task or game has been beaten on the indicated setting.

Unlock God Mode:
Beat the game on any difficulty.

Unlock Challenge of Hades:
Beat the game on any difficulty setting.

Unlock The Lost Levels:
Beat the Challenge of Hades.

Unlock Making of Attica:
Beat the game in God mode.

Unlock In-Game Movies:
Beat the game on any difficulty.

Unlock Inside Ready at Dawn Studios:
Beat the game on any difficulty setting.

Unlock Character Concept Art:
Beat the game on any difficulty setting.

Unlock Environments Concept Art:
Beat the Challenge of Hades.

Unlock Outtakes Concept Art:
Beat the game in God mode.