The best way to check an alternator is by load testing it with an instrumented carbon pile.
Most common way is to check the output voltage and assume the current produced is adequate if the voltage is "in range". With clamp on meters being pretty accurate these days (& cheap, considering it as test equipment), no reason NOT to make that measurement too.
Bet your gauge is a volt meter and not an amp meter - it helps to use an appropriate word for the quantity you are describing, especially with electrical things. You could also remove the alternator and take it to any good electric shop (marine or auto) or decent parts store and they should be able to test it on their bench.
Belt may be new but is it the correct one (size-wise)? That black powder indicates the belt is being chewed on; it may or may not be slipping. When we 'discovered' that stuff, the cause was corrosion induced roughness on the V contact faces of the pulley. Needless to say, even new belts didn't last long with the rough surfaces and would start to slip after ~ 10 hours of run time on the belt. Also worth checking is that the circulating pump spins freely.