Under the car

Begin by chocking one pair of wheels, loosening the wheelnuts and hub nut (drum brakes) on the other pair, disengaging the handbrake and taking the car out of gear, then raise the non-chocked side of the car and rest it on axle stands. Remove the road wheels and check the tyres for cuts, abrasions and bulges. Check the condition of the pads and examine the discs for scoring (disc brakes), back off the adjusters on drum brakes, remove the hub nuts and drums and check the shoes for wear, the drum for scoring, the backplate for oil or brake fluid contamination (hub seal or wheel cylinder seal kit needed) and (rear wheels) the handbrake components for free operation. With disc brakes, check for fluid leakage from the piston seals.
Check the dampers visually for signs of leakage and general condition. Check the brake hoses for bulges, cuts, abrasions or collapse. Gaiters on the drive shafts must be free from cuts, and will have to be replaced if they are damaged.
If you discover that any of these vital components have been swapped for worn-out alternatives following your inspection, then the best advice is to contact the vendor immediately and confront him/her with the facts, to contact your bank with a view to putting a stop on the cheque and, if the vendor denies any skullduggery, the trading standards office, police, any motoring organisation or owner's club to which you belong – anyone that you feel may be able to help.
Whilst the car is raised, take the opportunity to check the exhaust for signs of blowing, the visible sections of the fuel line for damage and the main battery/starter solenoid feed wire for signs of damage. Check any other wiring which runs under the car and inside the engine compartment. If you find any wires with damaged insulation then these need replacing before the car is used, and the battery earth should be disconnected there and then just to be sure that no wires can cause a short to earth.