A character profile is a detailed description of a fictional character's life and personality. They can be anything from highly elaborate to very basic. Its only goal is to help you understand who your characters are.
Before getting started writing any story, whether it be a short story or writing an entire book, these are the essential attributes a character profile helps you understand about your characters:
For relationships, it's best to focus on the ones that impact the story. For those, focus on the character's relationship to each other. Are they close or not talking. Or have they not seen each other in years. If so, why and why is that important to the overall story.
Relationships to consider: