Without disconnecting any wires, connect a DC-voltmeter (or multimeter) to the battery and check the voltage. Continue to start and accelerate the engine, while instantly watching the DC-voltmeter.
The volts will slightly increase when the battery is being charged. If not, the charging system is faulty. After running for a little while, the battery will be fully charged and the charge will reduce to just a trickle charge, so best to check the voltage increase immediately after starting.
Please note: without a proper battery connection there will be no charge output from the DC-charging system, so make sure that the battery is connected properly before starting.