Kenmore refrigerator not working.

Last night our Kenmore refrigerator stopped working and is no longer pushing cold air into the freezer (which in turn means that the fridge portion isn’t cold either). I have a french door fridge with a bottom freezer and I do not believe that the compressor is running. I pulled the panel on the back and know that the coils are clean and that the compressor fan is running. However, I don’t think the compressor is running but it was extremely hot. I also heard a click coming from the compressor area about once every 30s to 1minute.

I unplugged the refrigerator and pointed a desk fan at the compressor to cool it down. Once cooled, I then plugged the fridge in to see if it would start up. I put my hand on the compressor and it didn’t feel like it was running. I heard the clicking noise and after a minute or so I unplugged the fridge.

Any tips on how to repair this? I have read reviews on this site that point to the start relay being faulty.