Grew up in Chapel Hill and still live near by..some of my favorite spots.
BBQ: Allen and Son (a hole in the wall place but my favorite BBQ in the state for NC vinegar sauce style BBQ)
Homecooking: Mama Dips
Diner: Elmos in Cameron or Breadmens on Franklin
Dessert: Sugerland (haven't eaten dessert there but they made my groomsmen cake and it was unbelievable.)
General Food: Crooks Corner (Love their shrimp and grits), Top of the Hill (amazing location, good food but had more fun going to its bar after hours than eating at its restaurant), the Loop (moderate price, self seating, but awesome pizzas and milkshakes), 411 (good place but a little pricey for a good but not great meal)
Also definitly check out Durham. It's only 15-20 min away and it opens up a whole new set of fantastic restaurants like Magnolia Grill, Four Square, Revoultion etc.