An approach similar to that will work but this is the biggest hurdle you will have:
The USB Client port on the Galileo is CDC, not HID and I don't see how it can be changed from within Arduino.
So the connect to the PC will be over a Serial COM Port, not a mouse/keyboard emulation.
Maybe someone else knows if the Arduino 'Keyboard' library will work, or if there is another solution?
Or maybe USB CDC (Serial COM) is ok for you?