Congrats on getting in to grad school, and welcome to the Triangle!
There are several nice areas that could suit your needs.
The areas around Duke Forest are nice, with lots of grad students living in that neck of the woods. Deerfield Apartments or Colonial Village at Deerfield are good places to start. Reasonably easy access to both UNC and Duke/Durham Regional for your wife's job hunt.
The area around Southpoint Mall in South Durham is very safe and has tons of amenities like nearby shopping and access to the American Tobacco Trail. Newer apt complexes in that area are likely going to be above your target price range, but some of the older complexes like Pine Ridge Apartments or Sutton Place are likely in your range. Still nice and likely to be roach-free. :-) The Southpoint area is convenient to both UNC and Duke Medical Centers, plus there are a lot of outpatient clinics in the immediate area, so convenient for your wife.
You might also look at places in north Durham that would be in the vicinity of Durham Regional Hospital (and also very convenient to Duke itself.) I don't know apartments in that area very well, but I've heard good things about Woodstone at Croasdaile Farm and Croasdaile Crossings, but I don't have a sense of price on those.
Just FYI, you will likely pay a premium for a short lease, so your actual monthly rent will likely be higher than rates posted online. Duke also has a housing finder website, which gets some smaller, private renters if you aren't as interested in big apartment buildings.
Duke University | Student Affairs | Community Housing
Good luck!