Charge Arduino through Computer but connect to external circuit
The grounds would generally need to be common.

That said, if the motor side has its own power source it is possible to isolate the PC/Arduino portion of the system from the motor part of the system by connecting all the motor control signals through opto-couplers. This would permit the motor subsystem ground to be separate. Note that really cheap opto-couplers can limit the bandwidth of the signals going to the motors so if you are using high frequency PMW controls make sure to evaluate and select components accordingly.