Check the outside - There should be no gaps, or wavy panels. These could be signs the car has been in a crash and that the bodywork was poorly repaired.
Engine bay - It looks nice, but be wary of a very clean engine bay. It's usually a sign that it was cleaned for a reason.
Engine - Look at everything, and check that there are no loose, broken parts, or leaks. You should also listen to it. The engine should sound smooth and even.
Tires - Check out the tires for signs of use and sidewall damage. Tires should present even wear.
Mileage - The odometer is the first thing to look for. Do the math. Does the mileage make sense for how old the car is?
Steering wheel - Does it feel smooth? Can you hear any noises as you steer the wheel from side to side?
Electrical components - Check if every interior electric component is working, including the stereo or infotainment system.
Air conditioning - Climate control is very important in a vehicle. Make sure that both the air conditioning and heating are working.
Documents - Check out if the owner has all the documents, including the user manual.
Keys - You should always have two sets of keys. If the original owner only has one, take into account how much you'll have to spend to make a new set of keys.