first, it's great that you are thinking about this & asking these questions.
second, with the multimeter tool you are mentioning.... it cannot read fast changes in voltage or current, so you will need to be doing things in the time periods of lets say ~5seconds at a minimum. so message here, is that music is not a good option, it draws fast current peaks, and you will likely miss those and get inaccurate results.
how to test
the general concept would be the following..... you need a way to draw different (known) levels of current from your electrical system, for example..... 10A for 5 seconds, 20A for 5 seconds, 50A for 5 seconds, 100A for 5 seconds, etc. Then while you are pulling the current.... you can measure the voltage at the battery or alternator. You really need a high current, current probe for this in addition to your multimeter. otherwise you will not be able to determine what the current is......
how to interpret results
Once you collect some data, see what current level causes the voltage to go down to ~12.7V - 12.9V. Let's say that is maybe like ~80A current draw for 5 seconds? (i am making this up)
Once you hit that point, you know your alternator/electrical system is struggling to keep up, and that's how much current your stock electrical system can provide safely/reliably, without degrading life of battery/alternator, etc.
You would need to keep in mind also,..... other electrical loads in your car will need current, so thing like heaters, defoggers, headlights, etc. you need to understand how much current those items require as well and hopefully balance this with your amplifier's current draw requirements.
Hope that helps.