I always buy privately, but always do my homework first. What goes wrong with this model? Which models share the same toys and do they go wrong?
Simple things I'd check:
The oil level,
The oil inside the oil filler cap,
The current fluid levels,
That the tyres are all the same or at least in axle pairs,
Does the underside of the car look clean? an oil leak will soon show itself and likely won't be tidied underneath.
Do all the panels look the same colour under a high intensity bike light?
The panel gaps on every panel, are they consistent?
What condition are the wheels in, this is often a good sign as to how the car has been treated
EVERY switch in the cabin. Do the windows drop at the same rate?
Is the aircon cold? If not, don't accept the "it just needs a regass" excuse, get them to do it as it might be something significantly more expensive,
Have any of the windows been replaced? If so, by whom and for what reason?
Does it brake straight and true from 40mph with no hands on the wheel?
Does anything rattle on the test drive?
Has anything been repaired by the dealer? Why? When? is it warranty work and still covered.
There's more, but I've got to get back to my transfer report...
N.B. I am quite picky when it comes to cars....