When you turn your television on and get nothing but a clicking sound you could have almost anything wrong with it. The two most likely places to look are the high voltage section and the low voltage power supply. In order I will explain what to check to try to resolve the problem.
First, is the standby light blinking in some sort of pattern? If so, count the number of times it blinks between long pauses. If it is blinking in a patten, this is an error code, and can be helpful in diagnosing the problem. Depending on the manufacturer the blinking pattern will mean different things. Error code lists are available by googling "(manufacturer) tv error codes". Proceed from here based on the information provided by the error code list.
If your set has no blinking lights proceed by unpluging and looking at the mainboard for burned resistors or bulging electrolytic capacitors. Electrolytics look like little cans, with shiny ends and are generally black, grey or blue. They have a positive and negative lead so be careful when replacing to get the polarity correct. If they are bulging replace them. Also replace any bad resistors found. At this point it would be a good idea to inspect all solder joints on all boards, but especially the Low Voltage and High Voltage sections, and resolder anything which looks suspicious.
Finding no charred resistors or bulging caps, and after resoldering all connections, your unit still does not work, check the horizontal output transistor for failure. Using a DMM on diode check, check the transistor out of circuit for shorts/opens between any two leads. A good unit should have somewhere between 500 and 1000 ohms one way and infinity when the leads are reversed. Only two lead configurations will show conductivity. One set of leads should show infinite resistance both ways. If the transistor checks bad, replace it and the flyback. Check all resistors and diodes in the HV circuit and replace any suspicious units.
If the horizontal output transistor checks OK, check all semis in the LV power supply. Replace any bad units found and check all associated resistors and replace as necessary.
If both the HV and LV sections check OK, remove the circuit board from the CRT and try the unit. If it runs your CRT is bad and needs replaced. If it does not run your problem is elsewhere.
Working one section at a time, remove and isolate any remaining sections while attempting to turn unit on. If unit works with a particular section disconnected, your problem is in that section and you will need to do normal troubleshooting procedure, listed above, on the section in question.
If after all the above procedures are tried, and your unit still isn't working, you may have to replace an entire board. These days most boards are relatively inexpensive. The most likely board would be the LV power supply which usually contains the HV power supply also.
Any parts or boards needed can be ordered at http://www.encompass.com
Good luck with your troubleshooting and be careful while inside the set especially near the High Voltage second anode where it connects to the CRT.
Thanks for using FixYa.