I've just dug out the check list I wrote up for my brother when he went hunting for a car. Several of the points have already been covered but for what it's worth this is it:
Engine: Should start easily (preferably from cold) without any puff of blue smoke. Rev the engine and check for blue smoke when the throttle is released. Take off the oil cap and check for water in the oil (will show up as a creamy looking emulsion).
One of the weird things I tend to do is smell the oil - a burnt or sulphurous smell suggests it's been badly overheated. A smell of fuel suggests badly worn pistons/cylinder bores or possible head gasket problems.
Do the same with the coolant - look for droplets of oil floating in it or emulsion. Coolant is usually green or blue in older cars or red in newer ones. If it looks a rust colour it has been neglected. If there is a smell of fuel there is a problem. (Don't get coolant on your skin - it's not nice stuff)
Has the car had its cambelt changed at the recommended intervals? Cambelt failure generally does a lot of damage to the engine so you need to be able to trust it.
Has it been serviced at the recommended intervals?
Tyres/suspension: Check tyres for tread depth and whether they are worn evenly. If they aren't worn evenly it indicates the suspension alignment is out. This could be suspension wear or old damage (or Kwik-Fit have done the tracking

).
Are any of the wheels dented/show signs of kerb damage. On the test drive, is the steering wheel centralised when driving straight ahead? Does the car pull to one side? Does the car feel stable?
Brakes: If the discs are visible, what is the wear like? Are they badly grooved? Is there life left in the pads? (minimum thickness for most manufacturers is 3mm).
Are there any signs of fluid leaks at any of the wheels?
On test drive, does the car shudder under braking? (indicates warped discs or drums). Is the brake pedal firm? It should not sink to the floor or need to be "pumped up".
How far does the handbrake lever pull up? 3 to 7 clicks of the ratchet is the usual range specified.
Lights and Electrics: Do all the lights work? Do all the warning lights go out correctly? Are all the lamp lenses free from cracks/breaks and discolouration? Do all the accessories work? Try everything.
Under the bonnet - give the wiring a look over. If any wires have been joined with Scotchlocks it has been inexpertly repaired. If any of the wires that go into the wings have been joined with Scotchlocks the car may have had accident repairs.
Body: Check for rust all over but especially round wheel arches, sills, door bottoms, and inner wings. If you can, lift the carpets and check the area round the corners of the floorpan - if there are any horror stories they will be here. Significant rust by any suspension mount, strut or shock absorber is a bad sign - likewise with the areas round the seatbelt mounts. Walk away if you find these.
Get close and look along the panels for any ripples that may indicate repairs. Look under the carpet in the boot - a rippled floor panel means it's been rear ended.
Panel gaps should all be even (including the panel to bumper gaps).
Large chips or cracks in the area of the windscreen swept by the wipers is an MOT fail.
Paperwork: Do the chassis and engine numbers match the V5? Is the car the same colour as stated? Is the mileage consistent with the previous MOT certificates?
Edit: Missed out the gearbox

: Does it go into all gears easily? Are there any odd gearbox noises when driving? Does it jump out of gear?
Does the clutch work smoothly? Where is the biting point? Is there any clutch slip (put it into high gear and give it a bootful of throttle from a low speed).
I hope this is of some use.
