In the original setup the alternator on the end of the crank on the kickstart
side of the engine supplies alternating current to a rectifier which converts
the alternating current to direct current to charge the battery and power the
ignition and lights. To regulate the voltage only one pair of the three pairs
of coils in the alternator are working until the headlight is turned on - this
then switches in the other two pairs. The weakness in this system was the rectifier.
I remember them costing 17 shillings or so and being fairly easy to replace.
The above system isn't capable of supplying a modern 12v system with decent lights.
Both the rectifier and alternator need replacing with uprated modern equivalents.
How you test the modern equipment does very much depends on what you've got fitted
but you will need a decent multimeter to do it.
All I can suggest at the moment is to inspect the wiring in the charging cicuit
looking for broken or shorted wires.