Cash is king for getting a good deal.
Know the book value PLUS what they are selling for in your area. If you need an emissions test, can it pass. For $$X pay a trusted mechanic to give it a once over. The money invested up front could mean money saved down the road. Regarding carfax, unless an issue was reported, it will not show up on a CF report.
IF the seller say he/she does all their own work, have them show you the receipts for parts purchased. Oil changed at a jiffyoilcanrapid shop. Those places have records of their service work.
IMO, If there are no receipts, there was no work performed.

BTW: it's not a private purchase, it's a vehicle you purchased form one other than a dealer. Or as we say, i got it from the guy down the block