The flyback is the thing with the fat red wire coming out of it (and perhaps a couple of others going to the CRT board or it is near this component if your set has a separate tripler) and may have a couple of controls for focus and screen. It should have some exposed parts with a ferrite core about 1/2-3/4" diameter.
The filament of the CRT is the internal heater for each gun - it is what glows orange when the set is on. What has happened is that a part of the fine wire of the bad color's filament (assuming this is indeed your problem) has shorted to the cathode - the part that actually emits the electrons. Normally, the heater circuit is grounded or tied to a reference voltage so when it shorts to the cathode, the cathode voltage level is pulled to ground or this reference.
You will need some well insulated wire, fairly thick (say #18-22). Find a spot on the flyback where you can stick this around the core. Wrap two turns around the core and solder to the CRT filament pins after cutting the connections to the original filament source (scribe the traces on the board to break them). Make sure you do not accidentally disconnect anything else.
This winding should cause the filaments to glow about the same brightness as before but now isolated from ground. If they are too dim, put another turn on the flyback to boost the voltage as this will result in low emission, blooming, and possible damage to the cathodes after awhile. (Don't go overboard as you may blow the filament totally if you put too many turns on the core - you then toss the TV.)
Route the wires so that there is no chance of them getting near the high voltage or any sharp metal edges etc. Your picture quality may be a tad lower than it was before because of the added stray capacitance of the filament wiring being attached to the the (formerly bad) video signal, but hey, something is better than nothing.
If you are not inclined to build your own isolation transformers, kits are available.