Chapel Hill is definitely not more diverse than Durham (http://en.wikipedia.org/wiki/Chapel_Hill,_North_Carolina#Demographics vs http://en.wikipedia.org/wiki/Durham,_North_Carolina#Demographics) but I doubt you'd find much animosity toward mixed-race families there. It's very progressive, partly due to the presence of the university. I'm mixed race myself, and while I felt more comfortable in Durham, I certainly didn't feel uncomfortable in Chapel Hill.
Whether you live in Durham or Chapel Hill might be partially dependent on what kind of commute you're comfortable with, as well as what type of city you want to live in. Chapel Hill has more of the small town aesthetic, whereas Durham is more artsy (there's a revival of the downtown area going on right now -- lots of food trucks, repurposed warehouses, art galleries). I've lived in both and liked both.
Both Durham and Chapel Hill have lots of great child-oriented places. Check out the Museum of Life and Science in Durham (I loved that place growing up!). There's also the Botanical Gardens in Chapel Hill, Duke Gardens in Durham, Durham's food truck rodeos (lots of kids there), both cities have YMCAs.... Raleigh, nearby, also has the state-run museums of natural history, etc.