Hasankeyf is a village in Southeastern Anatolia, located on the banks of river Tigris.
In early August 2009, there was a military checkpoint about a few km south of Hasankeyf, on the road from Midyat/Mardin, with a commandant who can speak basic English. All they do is a simple ID check, though, so don't worry.
North towards Batman
Do the obvious — pass the bridge and start thumbing there. It takes half an hour at most to attract a ride. You may even be more lucky during weekend evenings as Hasankeyf (and the river) is some kind of magnet to spend the weekend for the people from Batman and the road is full of cars returning home during weekend evenings. However, it may not be such a great idea to hitch towards Batman when shadows start to grow longer, as you may stuck in Batman through the night and according to locals, that city is not very safe at night.
For camping—which is safe according to locals and is legal according to the commandant at the military checkpoint despite what some (Turkish) guidebooks say—, the northern banks of Tigris—those that the ancient ruins are not situated at—is better than the southern banks as it's quiter, greener, and much less crowded, except during weekends of course. Erect your tent when it's getting dark during such a time unless you want the local kids persistently test their English —consisting of “hello”, “what’s your name”, and “money”— on you.