I also would advise against it. Refrigeration compressors aren't designed to run with non-condensible gasses like air. You're also going to have oil problems, a refrigeration system is designed that the oil from the compressor travels around the system and eventually returns to the compressor. To use it as an air compressor you're going to need an oil separator and return system. Even then some of the oil will still end up in the receiver tank which will play havoc with the air tools and even more so if you ever use it for spray painting. Add this to the fact it will be running outside of its rated design suction and discharge pressures, it will have a short life span.
If you still want to go ahead it will wire as a CSCR motor so you'll need the start relay, the run cap, the start cap, a pressure switch, a safety switch and an on/off/isolator switch. I would also include the thermal Klixon plus any internal thermistors in the circuit wiring as token LRA protection.
If you can give the exact model number I'll give you a diagram.