In the Kelvin timeline they had a relationship, if brief, as referenced by Carol Marcus:
CAROL: Yes, you do. I'm a friend of Christine Chapel's.
KIRK: Christine, yes. How is she?
CAROL: She transferred to the outer frontier to be a nurse. She's much happier now.
KIRK: That's good.
CAROL: You have no idea who I'm talking about, do you?
KIRK: What are we doing in here?
CAROL: ls this shuttle prepped to fly?
Thanks. I'd forgotten that.When Kirk meets Carol Marcus in Star Trek Into Darkness, she's not inclined to like him because she was a friend of Christine Chapel's. Kirk replies, "Christine, yes. How is she?", but it quickly becomes clear that he doesn't even remember her. So presumably things didn't end well between them (especially if Chapel was Kirk's subordinate).
Chapel didn't appear in the movie or anything. It was just a throwaway mention. (She was also mentioned by name in ST09, IIRC.)