I've lived in Chapel Hill for about seven years. It's an amazing place to live! I've rented the entire time I've lived here, and in my experience, if you can find a house/condo/townhouse from a private landlord, it'll be a much happier experience for you than renting from any of the apartment complexes in town. Avoid Glen Lennox like the plague (it's in your price range, but trust me, it is not worth it). I suggest going on Zillow (or even Craigslist) and looking for someone who is renting out a property or a room somewhere. In my experience, because the demand for housing in the Triangle is so high, many of the apartment complexes in the area are overpriced, predatory, and horrible to their residents. Private landlords should also be more flexible about the length of a lease- all of mine have been fine with 12 months. A lot of apartment complexes will try to get you to do something longer.
Good luck, and let me know if you have any other questions!