apart from all the background checks, such as V5C, engine numbers, VIN numbers, service history, you also can check past MOT records on DVLA website for free with the mileage.
now down to the car
1, overall conditions, extra attention to the give-way signs such as excess wear on steering wheel, gas peddle, seats.
2, accident/repair signs?
3, timing belt/water pump changed? evidence of changing?
4, ball joints
5, uneven ware of any tyres?
6, start the engine when cold with bonnet open, listen to it, after it warms up shut the bonnet, you shouldn't be able to hear the engine running
etc.... I only got a few mins to response to this list, I am sure others will get you a much better list, will come back to update this when I got a few mins
Felix