Bounce the car on the front passenger wing car should go down come up and stop. If it carries on bouncing there could be suspension issues, you should try this all the way round but the back is a little harder to do so i would concentrate on the front as your handling could be gravely effected.
Tyres!!!! Tax and MOT!!!! If it doesn't have an MOT ask why don't take any **** like "Oh... i just have not had time" because anyone genuinely selling a car would ensure its road legal as it is illegal to sell a car in the knowledge that it will fail an MOT unless the seller makes it clear why it would not pass an MOT.
Make sure the engine starts promptly and idles smoothly at around 1000 RPM.
Listen for any worrying noises from the engine such as knocking or grinding noises. Tapping is to be expected but only a very slight noise.
Get in the car make sure you can change through the gear box with the clutch down and the engine running. Check that 1st and reverse are easily selectable. If you can test drive, pay attention when pulling off. Feel for vibration and/or knocking noises.
Check the brakes work! Check the Seatbelts work as these will save your life if they have to heh.
Think as Mork said the general rule of thumb is to keep your eyes and ears open, listen to what the seller is saying, are they actively trying to sell you the car or do they seem uninterested. I often find a genuine seller will inform you of any problems straight away and will appear very reasonable but just remember if its too good to be true if often is.
Stay sceptical as this will keep you and the seller on your toes.
Be sure to ask questions regarding the cars origin, previous owners and recent history(Whilst owned by the Seller).
Without meaning to be sexist in anyway, shape or form I would recommend a male accompanist as i have seen a lot of guys jargon bashing woman in a sale and generally being quite patronising and even condescending to women buyers.(I can assure you this is not the attitude of all males
)
Well that kept me out of trouble for 5 minutes hope any of it helps