Level 2 Masks:
Start Offline Skirmish mode, and play 1 round each of Smakahuna, Rodeo, and Boar Polo.
To bypass, Enter the following code in the options menu: icheat

Level 3 Masks:
Might be possible to Unlocked from Offline Skirmish mode. We still don't know the requirements to unlock it yet.
Originally you could unlock by Winning 3 Online Battles
To bypass, Enter the following code in the options menu: thereforeiam

Level 4 Masks:
These masks can be unlocked offline, but the method has yet to be determined.
Originally you could unlock by Winning 9 Online Battles
To bypass, Enter the following code in the options menu: soveryvery

Level 5 Masks:
The only way to unlock these masks was while playing Online.
Originally you could unlock by Winning 15 Online Battles
To bypass, Enter the following code in the options menu: weakandsad