I wanted to start a new thread on this, since there is so much good information that this subject requires. Let's collect as much info and tips as we can!
I'm starting with a general overview of the chamber reaming process that I have read. I know from my limited experience that the condition of the lathe is important. If you can't true up the lathe, then you don't need to be reaming a barrel on it!
The lathe requires a steady-rest to be used to support the barrel. That gives you two support points - both the head and the steady rest. You must true up the barrel so there is little, if any, runout. This is the most time-consuming part of the process. PREPARATION IS KEY! Only if the barrel is properly prepared can the reamer do its work.
Use plenty of cutting oil, and be sure to clear chips as often as possible. Low RPMs are important so the reamer doesn't dull or seize. Recommendations I've read are "up to " 200 RPM for the reamer.
If you are not sure that the outer diameter of the barrl is true, then use the dial indicator to measure against a tight-fitting plug in the bore. That will ensure the chamber is calibrated to the bore and not the outside barrel diameter. This may not be a necessary step.
With the barrel true, use a starter bit to cut a shallow recess in the barrel. This will help keep the cutter aligned when it starts to bite. Use either a roughing reamer or an under-sized drill bit to start the cut. Cut to a point before the chamber shoulder should begin. Then remove the rougher and use a finishing reamer for the final part of the cut.
Use either a headspace gauge or some other measuring device to measure the depth of the cut. Too deep and you'll have to remove metal from the breech to get it right. Too shallow, keep on cutting.
A new tool I've discovered (thank Bradrock!) is a "floating reamer holder" that lets a slightly out-of-spec tailstock do the work. It lets the reamer "float" by using either a spinning collar or a ball bearing for the axial thrust. The reamer is held in place by either a box wrench or a tap wrench.
Finally, you can polish the chamber with sand paper or a dremel polishing wheel, then with a metal polish like Flitz.
You guys out there with more experience and knowledge on this topic, please post more info, or corrections to my short explanation!