To charge the battery while the console is off, you would first need to have the console turned on. Once the console is on, plug in the charge cable and attach it to the controller. Once the cable is attached, you can turn off the console and the battery will continue to charge.
While the battery is being charged, a red LED will light up on the cable. Once the batteries have been fully charged, the LED will turn green.
Yes, once a full charge has been reached, the battery will stop charging.