What happens next?

How to buy a used car from a private party
Set aside a definite budget.
Private sellers expect to be paid by cash or check, in-full and upfront, and cannot provide financing. However, private sellers tend to be quite a bit cheaper than dealerships, making an all-at-once payment more beneficial.
Note, however, that consumer protection laws do not apply to private sellers. Check the kelley blue book value of any car you are considering.