1) Clara was raised by women, growing up in a little house with her mother Doreen and her Abuelita. Her father Lawrence died when Clara was four. Doreen was an ER nurse who went back to school to become a surgical technician. Her Abuelita worked with their church and volunteered at local soup kitchens. There were no men in Clara's life until her mother started dating a coworker named Ben when Clara was thirteen. The Uncle Ben jokes started very quickly thereafter, although he has yet to die in order to advance Clara's heroic origin story.
2) Clara frequently rattles off quips about her intellect, the first to remind others that she's smarter than they are. This isn't just for show, however, as she possesses a genius-level IQ of 145. She's also 19, and hasn't quite come to grips with the fact that bragging about how smart you are doesn't endear people to you.
3) Clara has few real hobbies. Her entire academic career was consumed by advanced classes, summer math camps, and after-school science programs. If it didn't win her a trophy or a scholarship, it wasn't worth doing. When she discovered women's flat-track roller derby during a summer abroad in Germany, Clara fell in love with the sport. She became the jammer of the Bear City Brawlers under the derby name Snarl Sagan. She's very proud of that pun.
4) Clara is an amateur cosplayer. There are time-worn and much-beloved Spock costumes in her closet that she and her mother began making for San Diego Comic Con, including a genderswap version based off of Uhura's costume. She still takes them out for conventions whenever the opportunity presents itself.
5) Clara has a collection of sketchpads full of costume designs that she and Hannah have come up with for the rest of the cast. Nobody else is nearly as excited about this as Clara and Hannah are.
6) Clara refuses to speak of JJ Abrams or his Star Trek movies. It didn't happen. She'd rather talk about Star Trek 5.