Get your local mechanic along and take the car out for a spin. Check for engine vibrations at high speeds, smooth transmission, and under/over steer. Check for the car's maintenance records to verify if the car was cared for.
Take the car to a garage and get it inspected by the mechanic.
This should take an hour at most, so it shouldn't be a time constraint to anyone. Besides the odd chance of uncovering a large fault, the mechanic could find a few minor glitches that will pull up your bargaining power.
Jot down the glitches the car has and bring them up while haggling on the price.
Also read: Credit cards: Beware of 'conditions apply'