Have a look at the general state of the engine. A dirty engine and surrounding area suggests that the car hasn't been well looked after and that servicing may have been neglected.
Conversely, a sparkling clean engine could have been steam cleaned to disguise problems such as oil leaks etc.
Before starting the engine remove the dipstick and check the colour of the oil. If it is very black the car has probably not been recently, or regularly, serviced.
Also check for beige "mayonnaise" on the dipstick, a possible symptom of head gasket leakage.
Check the quantity and colour of coolant. It should be the colour of antifreeze, not rusty red. An engine that has been run without antifreeze may have problems.
Listen to the engine starting up from cold. The oil light should go out soon after the engine starts - if it doesn't there may be engine wear.
Heavy rattling or knocking noises shortly after start-up could indicate wear of the crankshaft and big end bearings. Listen for clattering or light knocking noises from the top of the engine which indicate camshaft wear.
Turn on the ignition and open the throttle sharply. Check for black or blue smoke from the exhaust. Blue smoke comes from burning oil and shows engine wear, whereas black smoke is unburnt fuel and has many possible causes.
Check for smooth idling when warm.
When test driving check the engine does not misfire but pulls strongly and cleanly. Check there are no pinking sounds (i.e. metallic rattling sound that occurs when the throttle is open).
If the car does misfire and the engine has electronic fuel or ignition control only buy it if the misfire is put right first.
Keep an eye on the temperature gauge or warning light which may indicate overheating.
Finally check for oil leaks.