When playing a minor character, examine what you know of the canon world, and details surrounding not only your character, but the characters they are most frequently around. Examine the character's reactions to things and the character's behaviour. In the case of one of the characters I play, her sibling gets a decent amount of backstory and screentime, but my character is very minor with only a few spoken lines. However, since I know more about the sibling, I can make some reasonable headcanon for upbringing and I can note contrasts between her and her sibling's experience.

Also, never do anything based on fanon. It's fine to look at what other people have done with the character and look at what aspects of the character they develop, but make sure to canon review and make your own decisions on what you're going to do.