Hey!
I'm also training for the Great North Run, and for the past couple of years at Durham I have done various jogs during the summer term - usually I go jogging and a friend of mine takes his bike and I just try to keep up with him!
I'm unsure of where you're starting, but one route I do is:
Trevs --> Up the road that goes through Grey --> Past the Botanic Gardens --> Down the footpath through the forest (down the quite steep slope) --> Turn left and follow the footpath all the way to Maiden Castle --> Cross the main road by Maiden Castle, but keep following the forest footpath --> This should eventually take you by the river, keep following the footpath til you come around by the racecourse/rugby grounds/Durham Amateur Rowing Club --> Go along the riverside path/by the bandstand --> Over Baths bridge, turn left --> Follow the riverside path all the way to Prebends bridge --> Cross over Prebends bridge and head up --> Turn right at the roundabout and go uphill --> Turn left at the top of the hill onto Elvet Hill Road --> Back to Trevs
This is certainly around 5km I would say, and has a good mix of hills and flats
If you want me to put it on a map let me know - in fact we can use this thread to think up lots of routes.
Last year a friend of mine was training for the Edinburgh Marathon and he had a 10-mile route around Durham which he did about once a week - I'll try to find out what it was!
Otherwise, just grab a map of Durham from Google Maps or something and just think about where you can go, there are lots of various routes you can do if you know where all the footpaths are!