To select an alternate color, simply highlight your desired character and press one of the five buttons that don't correspond to "light attack". Pressing "light attack" simply chooses the default colors.

Each color matches up with each button, which I've denoted with single letters.
L - Light Attack (Default A on keyboard or A on 360 controller)
M - Medium Attack (Default S on keyboard or B on 360 controller)
H - Heavy Attack (Default D on keyboard or X on 360 controller)
A - Assist (Default Q on keyboard or Y on 360 controller)
P - Parry/Guard (Default W on keyboard or LB on 360 controller)
N - "No Action" (Default E on keyboard or RB on 360 controller)

Yui Kutuna
 
Haruka Kutuna
 
Lilith
 
Luna Himeki
 
Kurumi Mirumati
 
Saki Mitonoya
 
Kaede Hioh
 
Natalia Glinka
 
Eri Hasumi
 
Ayane Ikuse
 
Hilda Rize
*Hilda's colors might be very wrong for reasons explained in the next section.

• If anything on this list is wrong feel free to leave a comment and I'll try to fix it.
  
• I don't actually own the Hilda Rize dlc so I'm basing her colors off the sprites that are in the files for my version of the game. There's a pretty decent chance that they didn't mess with her at all from a graphic standpoint, so these may still be valid.
  
• Technically speaking any of the palettes for any character could be wrong seeing as I don't have the time to manually select every character and color combination. However, I tested enough where everything should be linked up perfectly unless a character randomly uses a different pattern for what order their palettes appear in the game files.
  
• Which means yes, Eri's "Assist" and "No Action" palettes ARE supposed to look that similar.
  
• I was originally going to include the colors for all assists too, but it turns out they have extra colors per character they can be paired with, which means that there could be over 50 unique colors per assist, and no way am I getting all of those.
  
• Player 1 and player 2 can't seem to pick the same palette in versus mode even if they play different characters, which makes the above note even stranger.
  
• Trivia: Various characters have "beta" palettes in their files that don't appear in-game. This includes Saki with light brown hair and dark brown dress, and Natalia with a darker grey outfit. Various character portraits in the files use Saki's old colors while Natalia's old colors can be seen on her character select icon.
  
• The dog in the guide's icon is in the files for every stage, but doesn't seem to appear in the game at all. Spooky.