Durham Courthouse is the most famous spot in NC so hit that up for some history. There are the granite ledges and a rail on the other side.
Down the street there is a brick road that has some pretty cool planters. If you are into wallies and wallrides I suggest you check that spot out.
There is also an outdoor skatepark on the northern end of downtown Durham, it gets crowded as FUCK though. But they can tell you about some of the more secret areas.
And if you got the time, you can hit up
Raleigh (around 20-30 minutes away) which has more spots, more cops and Dan Murphy sightings.
And the weather in NC sucks in late january/early february. It might not snow, but its rainy and cold (which is worse)