Here are your parts
Parts for Kenmore Refrigerator 25356293409 - AppliancePartsPros.comThe Sears parts site does have actual part breakdowns. Unfortunately they do not share these with other parts vendors.
You have to locate where the clicking is coming from.
The most likely pace is the compressor. It is tring to start but for some reason cannot so it overheats, trips the thermal protect and then retries starting when the protect cools down (resets).
Pull the unit out from the cabinets and remove the rear cover.
If the condenser/compressor is running then the compressor should be on.
You can usually hear it but also should be able you feel it vibrating.
If it is clicking then it could be the compressor or the start device.
Below is a link on how to check them.
Note that the compressor winding resistance check is not a definitive check. In other words it can tell you if a compressor is bad (re: burnt out winding) but does not tell you that the compressor is good as it may have winding to winding shorts or mechanical problems.
Other than the physical check of the start device there is not a good way to check it electrically. So if the compressor windings check out OK and the start device also looks/feels OK then the normal troubleshooting is to replace the compressor and hope that it is the problem.
How To Fix a Refrigerator - Appliance Repair Guide - ACME HOW TO.com