I would start with the simple stuff first. Check all air flows and air filters first. Are the blowers working? Do you have a multimeter and do you know how to use it? If air flow is good ( coils clean and no air flow obstruction) then we can move on to looking at the compressor. I would start by pulling an amp draw on the compressor, is it running? If not I would remove power and pull the plastic cover on the compressor there you will see three wires. the common (usually black ) it goes to a little round plastic over load and then to the compressor. It could be bad or weak you can check it for continuity if it's bad that might be the problem or part of it. Next check for shorts and grounds on the windings of the compressor. Take your multimeter and set it on ohms you will have three pins sticking out of the compressor ( you have to remove the wire's first, take a pic with your phone) common , start and run. Take a reading, put one probe on the common and touch run and start with the other you should get a reading (the resistance of the motor windings) no reading? Bad compressor. Now with one probe on common touch the other probe to bare copper lines coming out of the compressor you might have to scratch the probe on the copper to get a good connection if you get a reading your compressor is shorted to ground and the compressor is bad. If it passes these tests it could still be bad, it could be stuck this would show high amp draw and the compressor will be very hot. Now if you don't find any of these problems. Then you will need to look at the start and run components and don't forget about the t-stat ( I unusually jump it out at the unit if it then works you might have found your problem).
Now for the disclaimer: you do this at your own risk, if you aren't handy don't do it. Some of the tests require power to the unit, having an understanding of wiring diagrams will be helpful. I know that the people on this forum are handy but electricity is a different animal that can bite you and hurt the whole time it's killing you. Again I am willing to give advice but I will not take responsibility if you hurt yourself you have to use your judgment as to your abilities.