having tried my hand at appliance repair, your best case scenario is that the problem is the compressor start relay is defective. this is an easy, cheap fix. the worst case is that the refrigerant line has a crack in it somewhere. if this is the problem, you're best off taking it to a scrap yard.
iirc, with a bad start relay, you'll have some clicking of the compressor and it might get hot but it won't really run at all. if it is the refrigerant line, you'll have a more robust run from the compressor and you'll even here some activity in the lines, but it won't get cold. it might cool down a little bit, and some of the lines might be cool to the touch.
most start relays are simple little devices. find it, take it out, take it apart, and inside there is usually a ceramic disc. if you see some scorching on this disc, that was probably the problem. often you can just rotate the disc and the problem is solved for $0.
it could also be the thermostat or something else. if it is an auto-defrost model, there are some other potential problems. what is the model of the freezer?