Trouble Shooting Small Haier Refrigerator

I have a very small Haier refrigerator, the type you have in a small office or dorm room. It suddenly stopped working. The only thing I have access to troubleshoot is the compressor and relay.

The relay part is P6R8MC OR0404. The compressor is an LG NSA24LACG.

What is the best way to troubleshoot? Previously, I have just plugged in and it turned on and worked. When it stopped, it makes no noise at all. No start up noise and no heat at all from the back. It used to get warm if not vented properly in rear. Any suggestions?