It's really easy.
1. Disconnect the 3 bullet connectors.
2. Set your meter on ohms
3. Touch one lead from the meter to the engine case. I used the rear manifold to avoid paint getting in the way. It does not matter which lead goes where. With a lead touching the engine, take the other lead and touch each of the 3 yellow wires in turn. Make sure they are the wires coming from the stator, not the wires coming from the R/R. You should get no continuity between any of the wires and the engine. If you do, your stator is shot. If not, move onto the next test.
4. Set your meter to VAC. Start the engine and rev it to 3000 rpm. Now check the AC volts between all 3 of the yellow wires, 2 at the time. Do not let the wires touch each other or ground. You should get high 40 to low 50 AC volts at each wire on a stock stator. If you do, your stator is good.
5. Now with a fully charged battery showing 13.xx volts with the engine off, start the engine and rev it to 3000. With the meter set to DC volts, and the leads connected to the battery terminals, you should get over 14 volts. If you do, everything is ok. If the voltage does not go up, or drops (the lights draw power from the battery, and can make the voltage drop down to around 11 volts if the system is not charging) the system is not charging, but the stator is ok.
6. There are a lot of possible problems, but I'd bet on the R/R, as it is the most expensive part. But check for proper ground, check the R/R connector, check the headlight relay, etc. You can check the R/R, but as was said, it can show good and still not work. If nothing else is found, and the DC voltage at the battery connections is low and doesn't budge when the engine is revved, you can be pretty sure it is the R/R
A charging system that is not working will cause the bike to misfire badly and die at idle, even with a fully charged battery. Apparently on the Vulcan, the ignition system requires the charging system to be working for the bike to run right.