The only true way to test is to replace. I had bad solder connections on my board, resoldered but it still would not work so I bought new and wala its working.

To test:

Pin 31 is ground
pin 15 is B+ 12V when ignition is on
pin 87 is out to compressor.

KL ? Speed
87K ? speed
VL ? climate control
GM+ ? engine speed
Not sure which is which but I do know that you have a speed reference signal from your speed sensor in back of the compressor on 2 of those pins. One other is connected to your engine speed reference signal on mine its "TD" and the last is from your climate control module.

To test Kilma;
Remove and jump 87 and 15 and see if compressor comes on. If compressor not on check fuses, wiring and connections to compressor. If so then check signal from speed sensor should be 0.2~0.4Vac across the 2 wires coming from the sensor not ground. Check speed sensor. You also have a microswitch somewhere under air cleaner box connected to the throttle linkage, it is used to cut off compressor when you give full throttle.
If all checks ok I would open relay module and check for bad solder connections. This is a common problem.