I always use my multimeter to ohm test the compressor start relay.
Put your meter probes in the top 2 holes, and put your multimeter at the lowest ohms of resistance, it should read between 3-12 ohms. If you don't get a reading between 3-12 ohms, or you get no reading at all, then its bad.
If the compressor start relay tests good, that just leaves the compressor as the culprit.
Jake, you led me to my question: How much do you think are we talking about if the compressor needs replacing? Do you personally think it's something worthy to chase or it's cutting it close to just buying a new refrigerator? Personally, I don't like ditching things if it can be reasonably repaired to prolong its lifespan.