Your character is in a stable relationship of some description (if they are not in a relationship in the canon of your story, assume this hypothetically). Your character discovers that their partner has been cheating on them. How does your character react?
(Ideas: Does your character panic or assume it was a misunderstanding? Will their feelings change as they "cool off"? Will they storm out without a backwards glance or seek resolution? Would they forgive their partner?)
Does your character hold any religious or otherwise existential beliefs? If so, what inspired them to hold those beliefs? If not, why?
If you asked your character what his/her favorite color is, what would they say? Is there any reason they resonate with that particular color?
Does your character have any deep-seated fears? Do these fears motivate how your character acts in his/her day-to-day life?
(Ideas: Mentioning phobias are OK, but think beyond phobias. Does your character worry about not getting married someday, to the extent that he/she is often motivated to peruse romance or solidify his/her current partner? Does your character fear being taken advantage of and miss out on friends, information and other opportunities because they don't trust the words of others?)
Does your character have any particular pet peeves or annoyances? If so, is there any particular person, thing or event that caused your character to have this annoyance?
(Ideas: Think about your character's traits and how they fit into this question. For example, a person who is quite shy and inhibited might be most annoyed by loud, overbearing individuals.)
How does your character feel about his/her appearance? Has any part of their appearance been changed (e.g. tattoos, unusual hair colors, piercings)? If so, why did your character want to make that change?
What are your character's primary goals/dreams/ambitions? Why does your character want to attain this goal?
If your character fails miserably and loses sight of his/her goal (if you answered that your character doesn't have any huge goals/ambitions, assume this is a short-term or small goal), what happens next?
(Ideas: Will your character continue on with optimism? Will your character turn around and critically assess why things aren't working, seeking to change? Will your character break down and give up?)
When making friends or allies, what qualities does your character look for in others?
(Ideas: Think about how their traits fit into this question. If your character is quiet and serious, does he look for someone likeminded, or someone extroverted to bring him out of his shell?)
If your character was confronted with the news they are terminally ill with no hope of recovery, how would they react? (Note: If your character is immortal, invulnerable or a minor God/Goddess, assume something happens to end their immortality)
(Ideas: What is the initial reaction? What will the character feel later down the line? What sorts of things would they want to do before their time is up?)
Time to think from two different perspectives. Part one: If your character was asked what they thought their weaknesses/flaws were, what would they say?
Part two: If you as the creator were asked what you think your character's greatest weaknesses/flaws were, what would you say?
Part three: Are parts one and two congruent (how your character perceives him/herself is the same as what you know as his/her creator)?
Tag people here if you wish to do so.