The last 5 cars I have bought privately, I have bought the first one I looked at. This is partly luck, plus a bit of a rigid approach.
You can cover a great deal over the phone and on autotrader.
I always ask.....
Is the car, house, logbook and you all at the same address? The reason is 'normal' people buy a car and register it at Swansea and park it outside the house. If any of these are not there, then it is trouble. 'I'm selling it for my brother-in-law' spells trouble, because it is not normal, it is odd.
Remember, you are buying the person as much as the car. Rough people and rough cars go together. Is the guy a normal family man? Is his approach, similar to yours. Is he straight in doing business in a friendly, dignified manner.
I like all the paperwork present, old MOTs (who would not keep them) and as much service history and receipts as possible. A new MOT is also goodish. Autotrader is good as you can see so many pictures and save a wasted journey. Always always, do an HPI check. £20 is peace of mind, and helps when you sell it.
In the past, I would say walk away, there are plenty more out there, but with our cars, the pool is very small.
All the more reason not to get 'car fever'. Happened to me once. I bought a Vauxhall Cavalier because I just just wanted it. All caution, and the above went out the window. Got my money back, but it was a close call.