What horsepower on the Honda and what year? Where are you hooking up battery charger - to the battery posts? FYI - Honda marine alternators are very expensive, so you want to eliminate all else before buying one.
Here are some preliminary checks. All of these tests depend on you having a fresh, fully charged battery hooked up...
Quick test on alternator. Measure voltage on battery with engine off. It should be between 12.0 and 12.5. Start engine and rev to about 1000 rpm. Measure voltage on battery. It should be over 13. 13.5 is typical. Up to 14.5 is OK. If no change in battery voltage, first thoroughly check wiring from alternator to battery - pull and clean terminals and connectors on all power wires., especially the wire from the battery to the starter. Next check the output voltage off the alternator while engine is running at about 1000 rpm. It should be over 13 volts and not greater than 14.5 volts. If not, and depending on what size and year of your Honda, you may have an external voltage regulator coming off the alternator. That should be tested also.
It is possible that your battery cables are corroded inside and putting up too much resistance. The addition of the charger could be overcoming that resistance enough to allow a start. Use a set of jumper cables from your battery to the starter. Negative cable from battery negative to a good ground on the engine. Positive cable from battery positive to large red terminal on starter. If engine cranks you likely have a bad cable.
Again - all of this assumes that you are using a fresh, fully charged battery.