Check that the engine really is cold when you arrive to test it. (If it's just been running that might suggest that it can be difficult to start and the seller has had it running to 'warm it up').
Then turn the hazard lights on and leave them on (noting how bright they are). See below for an explanation.
With the engine still cold (and the car on a level surface), check that the oil on the dipstick is between the 'min' & 'max' marks. (If someone can't even be bothered to top the oil up before offering a car for sale, they've probably not done much else worthwhile to it either!). Check that there's no 'sludge' on the bottom of the dipstick (because that could indicate that the oil hasn't been changed for ages).
Then remove the oil filler cap and look at the surface of the oil. (Hint: Take a small torch; it makes it a lot easier!). It should be just as if you're looking at fresh oil still in its original container. If there's any 'mayonnaise' apparent, don't even consider buying the car (unless you want a bill for around £800 when the head gasket goes in the very near future!).
Turn the key to release the steering lock but don't start the engine. One of you should sit in the car and turn the steering wheel to-and-fro, while the other person watches their movements and those of the front wheels at the same time. If there's any 'play' (with the front wheels not moving at the very instant the steering wheel is turned) don't buy the car. (Steering faults can be expensive!).
Then carry out some checks on the condition of the seatbelts and anything else that doesn't require you to start the engine (such as looking for dents, scratches and signs of repair).
When you've exhausted everything you can do without starting the engine, take another look at the hazard lights. If they're dimmer than they were originally that means there's not much charge in the battery, suggesting it, or the alternator, has a fault. (You can then turn the hazard lights off).
Check that the car starts first time, preferably with one of you still standing alongside it to listen for any strange noises and to watch for smoke coming from the exhaust. (A bit of white water vapour is normal. Clouds of black smoke aren't!).
Then leave the engine running for a while, while you look underneath to see if any water or oil is dripping onto the ground.
When you take it for a test drive, use all your senses. Your eyes are for the road but your ears and your nose should concentrating on the car. Listen for rumbling or squealing sounds (suggesting perhaps a duff wheel bearing) and be alert for a smoky smell (which could be a brake binding, especially if there's a grinding sound as well) or one like electrical burning (which usually means that the clutch is about to fail).
Put the car into 5th gear at around 30mph and then try to increase speed. It should gradually build up. If it doesn't, suspect a clutch fault.
Check that you can find all the gears easily, including reverse! (I knew a very experienced car dealer who paid thousands of pounds for a Range Rover in apparently immaculate condition. It wasn't until he tried to park it on his forecourt that he found out there was no reverse gear!)
Remember to check that the heating (and, if relevant, the air conditioning) is fully working. Also check the radio.
OK. Those are just a few starting points. I'm sure that others will offer lots more but here are a few suggestions from the AA as well:
http://www.theaa.com/motoring_advice/car-buyers-guide/cbg_usedtips.html