When you complete the following tasks the corresponding character icons will be added to your save screen.

Shiki:
Complete the first week.

Joshua:
Complete the second week.

Beat:
Complete the third week.

Rhyme:
Have ALL items (Books, Clothing, Stickers).

Mr. Mew:
Have every pin mastered.

Hanekoma:
Collect ALL secret reports.

Rhyme (Noise Form):
Fight every noise at least once.