After the battery is charged, put a DC voltmeter across it. It should read about 12 volts. Then start it up and put the voltmeter across the battery again. It should now read about 13.5 or 14.0 volts. If it does the alternator is putting out and the battery is at fault or there is a short that is draining the battery.

To check for a short, turn everything off and put an ammeter in series with a battery cable. It should read very close to zero amps. If it reads more than about 50 milliAmps, you need to start disconnecting things while watching the ammeter. Start by pulling fuses one by one. When you pull one and the ammeter goes to zero, that's the draining circuit.

If with all the fuses out it still has current draw, start disconnecting circuits until you find the draw.

Good luck,