I would recommend the East Liberty area. There are some places where you can get relatively decent 2-3 bedrooms for well under $800/person. For example, JJLandCo has a somewhat decent 2 bedroom listed for $925 with AC and some utilities paid, really close to a direct bus stop to Pitt. Fairfield Apartments have some 2 bedroom and 2 bedroom townhome apartments for $1250-$1400, all in good units with lots of great amenities and walking distance to shopping and direct bus stops to Pitt. East Liberty Place also has some 2 bedroom availability, good amenities, shopping, bus access, etc., for a similar price. There are also a lot of houses available for rent by private renters within your price range with 2-3+ bedrooms.
The area has a bad history but has gotten a lot better. I have lived in the area for about 8-ish years, got my PhD from Pitt, been teaching post-PhD in Pittsburgh for a couple of years, and have had no issues and have seen vast improvements. Never experienced any kind of crime or violence.