With the understanding that you know what fits you (pay attention to geometry if you don't already), at a bare minimum:
Go through all of the gears to make sure they shift smoothly
Make sure the brakes work properly and without noise. Factor in worn pads and rotors/rim braking surfaces into your offer.
Make sure bearings are smooth and adjusted.
Look for rust and cracks, and any signs of abuse or neglect. This includes the frame, drivetrain, cables, and all fasteners.
Check for uneven spoke tension by plucking them and listening for similar pitches. Wildly different pitches could mean a damaged rim or improperly serviced wheels.
If you think you can fix any of the above, at least factor it into your offer for the time, effort, and risk.
Things I have been burned by:
Worn hubs. someone loosened the cones to make them spin smoothly, but they were crunchy when adjusted.
Worn bottom bracket. Again, they loosened it and I didn't notice the slight play during the test ride.
Cracked (carbon fiber) seatpost. I would have never thought to pull the seatpost, but got home and took things apart to give thorough maintenance and found a crack running from bottom to about 1.5" up.
Cut/shortened seat tube. A thick seatpost collar was hiding the angled cut, presumably to allow a shorter seat height (stoker on a tandem). Again, would never have though to remove it and check.
__________________
It's like riding a bicycle