It's probably best to take somebody who knows at least a little about cars and has bought a few before with you, if you know someone that is.
Some basic checks to do are:
Touch the bonnet of the car and make sure it's not warm (unless the guy has just got home from work) - a cold start issue is easy to hide by taking the car out for ten minutes before hand
check the oil level before you start the car, if the owner claims it's been serviced a week ago and the oil is horrible and black he's lying
Check all other fluid levels - anybody selling a decent car will make sure they are all at the required level.
Undo the oil cap, check underneath for a thick white mayo type substance, a little may just be an indicator that the car has done short journeys, especially just coming out of winter time. A thick coating hints at head gasket failure.
Check the block itself, is it clean enough without dried in oil/coolant stains?
Check the coolant hoses for white or green stains too - this indicates a leak or previous leak
Open the expansion tank and look at the coolant, make sure it looks relatively clean i.e no oily deposits and it shouldn't look in any way brown / thick
Lean on each corner of the car individually, check the side just levels itself off properly - if it bounces up and down the shock is buggered.
Turn the front wheels, if you hear any sort of clunking/banging noise and have a look at the bottom of each spring, where it seats - you should be able to see the last coil and it'll have a clean finish, if it looks sheared it is. Springs are not expensive to sort, but you should do the front two or back two at a time
Start the car, let it idle for a few minutes and have a listen to it, a petrol motor shouldn't sound in any way rough - if it's a diesel you'll have a harder time listening for anything that doesn't sound right.
Take a drive, everythign should still be smooth, no excessively high biting point, no overly heavy clutch (depending on the car), gearchange may be a bit sticky when cold, should free up once you've been on the move for a bit. A slightly notchy gearchange isn't a worry and on most older cars can be adjusted, it shouldn't crunch into gears under any circumstances though
Park up on a slight incline, put the car into 4th and try to pull away, if the revs rise and you don't go anywhere the clutch is going - or if the car manages to pull away this is an early sign of clutch problems too. You can also stick the car into 4th at say 30mph and accelerate hard to 60/70, it should pull evenly, no revs rising without speed rising.
If the car knocks over bumps in the road it usually means suspension components, can be arb bushes, droplinks etc - not really expensive to sort in most cars. If coupled with this it feels a bit like you're going sailing when cornering it's most likely the arb bushes
Hesitation when accelerating can point to MAF/HT lead / Coilpack or other sensor issues. A bit of common sense coupled with most obd code readers can pinpoint these reasonably well.
Check the abs light flashes when the key is turned, and make sure it doesn't stay on - probably wont be an issue, usually just a wheel speed sensor but i'd be asking why the seller didn't sort this already given they're so cheap.
Again check the engine management light flashes at start up then stays off. If it's on throughout the drive you'll need a code reader to check what the issue is - normally either the cat or lambda sensors will cause the light to flash but no appreciable change in drive.
When the car is warm, check the heating works and make sure it varies between hot and cold when asked.
Make sure the various wearing bits of the car match up with its claimed mileage - if it says its on 50k but has a smooth steering wheel and worn through bolsters then it's lying. Not an issue for a car with 150k to have these, but I'd be happier knowing nothing was being hidden from me
Can't really think of much esle off the top of my head, hope this helps a bit.