For erratic on/off problems, gently tapping on the relay when the problem occurs will confirm that the relay is at fault - if the set then switches on or off properly, it's almost certainly the relay and replacing it will fix the problem. But double check its solder connections to make sure it isn't a simple bad connection to the relay or in its vicinity.
So what exactly is the purpose of such a power relay in a TV set? Why doesn't the power switch on the TV just apply power directly instead of through a relay?
The usual reason for a relay instead of a knob switch is to permit a remote control to turn power on and off. If your TV does not have a remote, then it is simply the same chassis minus 24 cents worth of circuitry to do the remote function. Isn't marketing wonderful?
The only unknown is the coil voltage. It is probably somewhere in the 6 to 12 volt range. You should be able to measure this on the coil terminals in operation. It will be a DC coil.
However, the relay controls the 125 VAC (or 220 VAC) which you should treat with respect - it is a lot more dangerous than the 25kV+ on the CRT!
Almost certainly, the relay will have 4 connections - 2 for power and 2 for the coil. If it is not marked then, it should be pretty easy to locate the power connection. One end will go to stuff near the AC line and the other end will go to the rectifier or maybe a fusable resistor or something like that. These will likely be beefier than the coil connections which will go between a transistor and GND or some low voltage, or maybe directly into a big microcontroller chip.
However, a few sets use a latching relay - separate coils (sharing a common connection) to 'set' and 'reset' the relay - like a flip flop. In such a case, the controller knows which state the TV is in (on or off) and sens the appropriate pulse to the relay to change state. And, there will be NO voltage on the coils of a latching relay except when turning the set on or off.
Of course, the best thing would be to get the schematic. Some big public libraries carry the Sams' photofact series for TVs and VCRs. If not, take 10 minutes and trace it. You should be able to get far enough to determine the relay connections.
Once you are sure of the AC connections - measure across them while it is off and also while it is on. While off, you should get 110 to 125 VAC. While on and working - 0. While on and not working either 110 to 125 VAC if the relay is not pulling in or 0 if it is and the problem is elsewhere. We can deal with the latter case if needed later on. Note the even if the relay contacts are not working, the problem could still be in the control circuitry not providing the correct coil voltage/current, though not likely.
It may be expensive and/or difficult to obtain an exact replacement, but these are pretty vanilla flavored as relays go. Any good electronics distributor should be able to supply a suitable electrical replacement though you may need to be creative in mounting it.