While the appraisal you can do is significantly reduced from the kind of appraisal done by a technician, it still will keep you from getting a piano that is likely to need more repairs than you want to support. If you are looking for a technician in your area, check with friends or co-workers who have pianos. Ask them about the technician who takes care of their piano.
Word of mouth is always best.
Bring a flashlight and a toothbrush. Initial things to check are that all the keys play. Open the top and check that there are no hammers missing. It has a double row of little flat-topped pins with the strings running across it diagonally. Check around these little pins for cracks.