Price
Low mileage
Services records if available

Car facts like:
Number of previous owners
Accident history if any
Registration lapses if any

Note there are services for this.

Purpose of the vehicle, so what type.

Low miles when buying a used car, and I check out Consumer Reports at the library to see which car will typically have the least repairs.

Engine condition and age. I have no problem buying older cars but I won't pay for sentimental value

MPG, comfortability (something that would be nice to spend hours in if I had to), space...

Low Kms.
Service record

50k or less miles and a smooth ride

a place that sells cars.

I usually just kick the mud tires.