For starters, walk around the vehicle looking closely at the lines of the vehicle. Check to ensure spacing between body panels is the same and that the paint matches evenly. These are tell-tale signs of an accident. Next, open all the doors, hood, and trunk. Look on the inside of the door panels, inside the trunk, and down inside the engine compartment for paint lines, replaced parts, or overspray from painting. These are also signs of an accident.
While in the driver's seat, start and stop the engine more than once with the ignition key. Listen for any rough starting, rough idling, vibrations, other unusual noises or run-on after the engine is turned off. Anything out of the ordinary should be documented. If you live in an area that has varying climates during the day, ensure you try this during the most extreme time frame of the day. Once a list of symptoms is compiled, give it to your trusted mechanic.
When the car is running, move the gear shifter from park to drive, neutral, and reverse. Listen for any abnormal noises. If possible it is always best to bring someone with you who can listen from outside the car while you are testing it inside.
Slowly move the car and apply the brakes, listen for noise. You can also visually inspect the brakes and rotors to see if repairs may be needed. Inspect the tires to ensure tread depth and no visible signs of wear and tear or dry rot.
Lastly, drive the car at slow and fast speeds, ensuring you make turns to the left and right as you listen for abnormal noises.
If the radio is on, turn it off... If the person showing you the car turns it on to show you that, again, turn it off and check that separately when you are not testing the reliability of the vehicle.
Once you've assessed the pros and cons of the vehicle and see if it's worth investing your time and effort into. If repair costs outweigh the bargain price being offered, don't invest.