This is simple. To verify that the clutch is Ok hot wire it. After this test is OK, bypass the low pressure swith by joining the two terminals with a conductor. If this activates the clutch, the problem is there. Check if your system is at least 70 PSI. So, on the other hand the checking resulted negative, check the voltage on the terminals of that connector with the AC button "on". One of these terminal need to have 12V. If not, you need to go to the fuse box or the switch and continue checking. Let me know how you do.
(Make sure that you are testing the low pressure switch and not the high pressure).

Good luck!

I just converted my A/C over from R12 to R134A. I recharged with one can of R134A to about 70lbs system pressure. However, the compressor clutch won't engage. I can force it be engage manually by hot wiring with 12V. I've though about the low pressure switch but it was working fine before conversion. Any help would be appreciated.