Body damage isn’t always apparent. I always go to the back end of the vehicle and look down the right side and then down the left side, paying close attention to the lines of the vehicle. You’d be amazed at how many used cars you can find that look fine, but have lines that don’t match on the left and right (indicates the thing has been whacked pretty good and repaired).
Other things I look for:
Open the radiator cap (when cool!) and look for signs of oil.
Pull out the dipstick and look for signs of things other than oil.
Look at the engine for signs of oil leaks. Also, of you see a really clean spot near any sort of fuel line it probably means that the fuel line or component was leaking. Gasoline will dissolve oil leaving a clean spot.
Check the tailpipe for excessive black soot buildup (indicates that the car is burning oil, which means the rings in the engine are probably shot).
Excessive smoke out the tailpipe is bad. Look for white smoke (antifreeze) or black smoke (oil).
Check under the front and rear bumper for signs of repair.
Listen for popping noises when turning (bad CV joints).
See how smooth the braking is. Warped or chewed rotors will pull excessively towards that side of the car.
If the vehicle is a stick, drive up a very steep hill and try to get the clutch to slip. Generally shift through all of the gears to make sure they all work properly.
If it’s an automatic, make sure it shifts smoothly under a variety of conditions.
Use your nose. If the car smells musty in any way, run, don’t walk, away from it. Chances are it has been in a flood and it’s going to cause you no end of grief.
Check the tires for signs of uneven wear. Tires that are worn more on the inside or outside at the very least indicate an alignment problem. There are also some more serious problems that will prevent a car from being properly aligned.
Check the heater and air conditioner. Make sure that when you turn it on defrost that the air comes out through the right holes.
While taking it for a test drive, make sure you run over some potholes or bumps. You are looking for signs of something wrong in the suspension.
A lot of mechanics will take a look at a car for a reasonable fee. Having an experienced mechanic take a look at it is a good idea if you don’t have a lot of mechanical knowledge, or even if you do have a lot of mechanical knowledge and just want a second opinion.