That's a problem with your power supply, specifically how power is getting into the TV. Without the thing in front of me, I can't tell you exactly what has gone wrong.
First, remove the case of the TV so that the workings are exposed. Then turn the power on, and listen for roughly where the clicking is coming from. DO NOT put your head near it to try and find the exact location- you may shock yourself or cause a short circuit. Once you have identified roughly where the noise is coming from, turn off the TV and >>unplug<< it.
Next, inspect the workings. You're looking for anything that seems out of order: scorch marks around a particular component or connection, a component with a broken connection to the circuit board, a snapped wire, etc. From the sound of it, it would be a capacitor that has failed, although I can't be precisely sure. Look around where the plug leads into the TV.
Identify the component that has failed and hopefully the markings on it will still be legible, they will give you information on what you need to replace it. If you don't know what they mean, just bring it into a Radio Shack and look for one with the same markings and numbers on it. It need not necessarily be the same size.
The other possibility, and the more likely one in my experience, is that the power supply/adapter has failed. Is the adapter inside the TV or outside of the TV along the cable or on the plug? If it's external, just go online and order a replacement, it shouldn't be difficult to find one. If it's internal, just read the information listed on the part and search it online, and order that. You should be able to find one at a warehouse site or order direct from the manufacturer.
You may very well need to do some soldering.
