what amp are you buying?
check for pops and cracks while playing. also for any strange motorboating sounds. both are signs of bad filter capacitors in the power supply. these can be replaced though, and usually do need to be replaced after 10-20 years.
a very obvious check is whether there is any smoke (!) coming from the amp, which means that the amp has a bigger problem and should not be bought unless you're skilled enough to repair it yourself.
other than that, there isn't much. grounding is usually not an issue within the amp, a lot more depends on how your mains is grounded. if you REALLY want to be sure about the grounding WITHIN the amp, here is a simple test:
1. set your digital multimeter to continuity test
2. put one lead on the earthing pin on the power cable, the other lead on any exposed part of metal chassis of the amp, or perhaps the outer part of the input jack. it should beep.
3. to be even more anal, set to ohms measurement and measure the resistance of the ground between those points. you should get a maximum of 0.3 ohms (note: to get this reading, you will need a good, expensive multimeter like a fluke or something - cheaper multimeters are very inaccurate when things go less than 10 ohms or so).
other than that, can't really think of anything. make sure the speaker is plugged in or attached before starting the amp.