Dr. Mitchell is Beca's dad. Dr. Mitchell is married to Sheila. He teaches comparative literature at Barden University. It's mentioned throughout the movie that he disapproves of her dream of being a DJ and wants her to experience college.
It is implied that he and Beca have a strained relationship. He made a deal with her for her to join a club and after a year if she doesn't like college she can go to LA and pursue her dream of being a DJ.