ok, let the battery charge overnight and all day tomorrow.
Set your meter to 20v DC, ensure the leads are in (red)VOLT and (black)COM.
Touch the red lead to POSITIVE battery post, black lead to NEGATIVE battery post.
Tomorrow night take it off the charger take an immediate reading, it will maybe about 12.6, 12.7, 12.8 somewhere around there. If its down nearer 12.0 then that battery is finished.
***uming the battery is ok,, stick it in the car, and turn your headlights on for about 2-3mins but don't start the engine. Now take a reading here, again if its away down to 12.0 then battery is finished.
If that is all ok, start your engine and take a reading again, this should be above 14v. If its is lower and you rev the engine and the voltmeter fluctuates all over the place (like the headlight test) then its the alternator.