When you talk to a Muse, you'll get a sense of the items they're interested in and the outfits they want to try. Keep making Lookits that suit their tastes, and you'll slowly but surely become good friends.

Look out for special events
When you make Lookits, you'll earn hearts that show how close you are with a Character Muse. As the level increases, an "!" icon will appear over that Muse. Talk to them to raise your Affinity or receive items and special poses.