My frigidaire is not getting cold. Is there a way to troubleshoot to detemine if it's the compressor gone bad?

Good day,
Here is a basic checklist any qualified tech would use to isolate why a refrigerator won't cool.
Items that will cause a refrigerator not to cool.....
1. Defective thermostat
2. Defective freezer fan motor
3. Defective defrost timer or electronic defrost control.
4. Defective defrost thermostat
5. Defective defrost main heater.
6. Defective condenser fan motor
6. Defective compressor overload protector
7. Defective compressor start relay
8. Defective compressor start/run capacitor (some models only)
9. Defective compressor
10. Restriction in refrigeration sealed system
11. Loss of refrigerant (leak)
12. Excessively dirty condenser
13. Broken wires, bad wire harness connectors, ect.
14. Defective fan motor switch (some models only)

As you can see, there are multiple reasons a refrigerator won't cool, or cool properly.
To zero in on the compressor might take you away from the true non-cooling reason.

However. The compressor might have a defective relay or thermal overload that would prevent it from starting or running. Theses items can be removed from the compressor and taken to any appliance parts houses. Most all would be glad to test them for you.
As well, the compressor could have a bad valve internally that would prevent the compressor from pumping refrigerant, or there could be a restriction in the refrigeration lines preventing the refrigerant from circulating properly, or the windings in the compressor could be burned out or shorted, or lastly, the system could have developed a leak. All of the above compressor related issues, except for the first one could only be checked by a qualified tech.
I would treat the problem as a whole with the compressor being one of the potential issues.

Not to throw a monkey wrench into this but a refrigerator is like a car.
9 out of 10 times the problem and solution are obvious. It's that last 10% that take all the training and experience.