No Picture / Clicking Noise

Unfortunately, your problem is another one of thousands; I spend half of my time here pasting in the following explanation:
This problem is with fair certainty the power supply.
The clicking you hear is a relay that repeatedly connects, then disconnects the AC to prevent fire or further damage. 
 
If it just fails to turn on, the main power supply is probably at fault and without some technical background and a couple of test instruments, you will not likely be able to do the repair yourself. 

Modern power supplies are designed to shut down if the current drawn exceeds the design level, which indicates that something the supply services or the supply itself has died.
If you are adventurous, you might pull the plug on the set, allow the set to sit overnight, gain access to the innards, and with good light, inspect any boards inside.
If you see one that has few ICs but many more larger discrete parts, this will be the power supply.
Again, before touching anything inside, especially on that board, allow at least several hours to be safe since the larger capacitors can hold a painful voltage charge for some hours.  
You are looking for components called electrolytic capacitors that are almost always cylindrical and mostly installed upright at 90 degrees to the board with leads passing through to the solder side.
This same type of component in smaller dimensions is still used in a horizontal package with leads bent down and passing though holes to the solder side.
The latter are becoming more rare since they don't lend themselves well to robot assembly.
Many failed caps (not all) will show signs of pregnancy when they fail, bulging unnaturally at the top when compared with others. Now and then, there may be traces of a crystalline deposit around the end where the seal failed from internal pressure.      
These will have values listed on them in uFd & VDC and sometimes, a plus/minus number lying about the precision.
Some also have a date code (rarer) that will look like four digits:
 
2403 = 24th week of 2003
 
Most electronics suppliers have a stock of the various values but if they have a date code at all, try to get only those made before 2002 or after 2006.
 
The larger caps will probably be OK since the failure is likely related to functions other than brute-force filtering. A pretty good 'rule of thumb' is to replace any caps you see bulging of any value but especially those that are 100 uFd or less.
 
If you see signs of overheating such as discoloration of the board material, it may not be repairable but if you can find a part number on the board, you might be able to buy the power supply and install it yourself.
If there are no signs of heat damage to the board, there may be companies offering to repair it as a subassembly instead of repairing the whole set which could be much more expensive.
If you choose to replace it or have it done, the power supply still may not be at fault since most are capable of switching on and off if a downstream problem is causing an overload.
 
If you choose to replace failed caps yourself, you will need a quality soldering iron with a small, preferably iron-plated tip, rosin core solder and a sponge which when wetted is used to frequently wipe oxidized solder from the tip this should be kept bright and clean and fresh tinning will keep it that way.
 
You should also buy some solder 'wick' with the iron; this is used to place on the solder you wish to remove and then heated with the iron. Properly used, the wick will absorb nearly all of the solder from the lands from which you wish to remove a component.