I'd start tracing backwards - next up from the compressor should the relay, right? Can you swap a known good one? Or put a meter across pins 85 and 86 on the relay.

It definitely looks like the relay is closing to engage the compressor, now you've got to figure if something is telling it to; e.g. a short or something else; or if the relay is failing and it's just doing it on it's own.