The barrel has to be removed , thread pitch determined and 1/8 less than one revolution must be removed from the barrel shoulder. You must cut it a few thousands short of top dead center so that the new shoulder compress fits the frame barrel hole into top dead center so it will be tight and will stay indexed.
Use a real barrel vice and flat bar frame clamp set up to take the barrel off so as not to warp the frame if seated real tight. A wood stick through the cylinder port is bad practice and asking for trouble.
Now the barrel cylinder gap needs to be determined and that amount removed from the breech end of the barrel. It can be done very well with a file if a person is patient but a lathe is much easier and faster.
I usually don't mess with the barrel cylinder gap unless over .008 then it's time to set it back. I say barrel cylinder gap because end shake it actually a term used for cylinders with a bushing on the front. End shake includes barrel cylinder gap but is a distinctly different measurement and is generally about .001 to .0015 . Most factory guns, cartridge or percussion, come with a barrel cylinder gap any where from .004 to .010 generally. Dan Wesson's were very tight factory set at .002.
Now the forcing cones needs to be checked and possibly re-cut to proper depth. We have gauges for cartridge guns but cap and ball revolvers usually do very well with ball or bullet when cut to half a ball diameter deep.
We use special forcing cone reamers on center bushings on a shaft turned by hand from the muzzle. The cone is first cut then lapped with a brass slug and lapping compound.
Do not round off the inside corner edge of a re-cut forcing cone only break the sharp edge to discourage lateral gas escapement.
Some times the loading lever and latch needs modifying depending on how much set back was required.
This stuff is a gunsmiths bread and butter work !