This is one of my "great white buffalo" projects. There is a lot of good info on the internet, including pictures, from folks who have installed a conversion cylinder as well as straight up conversions of cap 'n' ball revos.
Least expense is to buy a replica "conversion" imported from Italy as a firearm. Second least expense is to convert using a conversion cylinder. Having a gunsmith make a conversion would be very expensive, and the 'smith would need to "manufacture" a firearm in the process. John Gren in WA used to do this and I actually sent a gun for conversion which he never got to before he got really sick. I actually got the gun back
but was out my deposit
The conversion cylinder in a Colt style revo really needs some additional work to the gun to match a period conversion, such as a loading cutout in the frame. So you don't necessarily bypass all machine work with one of those.
Anvil Enterprises used to sell plans to convert an 1851 Navy; these could be adapted to any similar open top Colt C & B pretty easy. I do not believe they are still around, but I'm not sure and maybe there are other similar plans out there.
There was an excellent book on revo cartridge conversions a while back (R Bruce McDowell ??) that is now super expensive. It had a lot of good detail on the real historical conversions, and explained the whys and whynots of the originals. It also showcased modern replicas of conversions on various other style revos, such as Rogers & Spencer replicas.
Back in the day these conversions were fairly common and often poorly executed by blacksmiths.
To get into the meat of your original question, you could perform all the steps just like they did back then. Lots of these were done by people with simpler tools than available today, but you really need a lathe and a mill.
The 44 cylinder Colt (or repo) has way too little meat to make a safe conversion. I have handled many of the originals and almost all have cracked cylinder walls and even little open windows where the charge holes blew thru the outher cylinder wall. Check out how the cylinder is rebated (smaller diameter in back). Look from the front; the nipple holes are not center line with the part you load powder and ball, and the two parts of the hole are different inside diameter. There is a real dimensional problem and these really require a new cylinder and frame modification.
The 36 cal Colts are much better to work with due to the lack of cylinder rebate and stepped frame. A conversion on the original cylinder can leave the locking notches in the cylinder alone.
The cylinders are also short once you turn off the part where the nipples screw in. The breech ring mentioned in a previous post fills the space where you remove the back part of the cylinder (not removed all the way to the center, just down to the part where the ratchets are located). This limits cartridge OAL.
The replicas are made of softer steel than modern cartridge guns. You need to be really careful and keep this in mind. On the other hand, the originals were iron. I would make my breech ring from something like 4140 or better and also sleeve the barrel and charge holes with similar. I'd also go with a weaker cartridge (32 or 38 S&W maybe) and load a BP substitute.
Lots of problems, but a really meaty problem and I think a great project. If you do not have access to a lathe (and know how to run it) you will have a low probability of any success. But don't be too discouraged, you can learn the necessary skills if you are determined. This is a project that you can accomplish.