1) Stuck or "hot" pixels. Shoot a shot at lowest normal ISO and at the highest rated ISO. View at 100% and look for little crosses. Assuming it is a Nikon, shoot both shots raw and run them through the free utility Pixelfixer http://www.pixelfixer.org
It can find stuck pixels in a NEF in a matter of seconds. Used it to get my D300 repaired before completing the purchase.
2) Check that AF works correctly.
3) Shoot a bright scene and a medium scene using Aperture or Shutter priority and ensure that the metering is working correctly.
4) Put an external flash in the hotshoe and take a shot to ensure the hotshoe is working AND that the flash is syncing with the camera.
5) If it has a pop-up flash, make sure it pops up and flashes.
6) Check the battery condition and life left if possible.
7) Shoot a shot at minimum aperture and look for dust spots on the sensor.
#1 though #5 are critical.
#6 may mean a bit more cost for a replacement battery.
#7 usually just requires a simple cleaning, but may get you a few $ off the asking price.