I tried to use a HC-05 bluetooth module with an Arduino Due through TX0/RX0 (Serial 1) but i doesnt seem to work. I can upload sketches to Arduino Mega but not to an Arduino Due. Arduino IDE doesnt find the Arduino, but I can open the COM port through serial monitor of the bluetooth module. But nothing is showing up in the Serial Monitor.
The Arduino Due doesn't use the serial inteface for sketch upload as the Mega does but has a direct USB connection to the main processor. You'll need a different boot loader if you want to upload sketches using the serial interface.
Please excuse, my answer was not completely correct. The Due uses the first serial interface to upload sketches if the programming USB port is used. But to program the Due that way you have to erase it's flash first by setting a specific pin and resetting it. The ROM then activates the boot loader. I don't see a possibility to activate these two pins using the HC-05, it may be possible by changing the firmware of the HC-05 and soldering some wires to the daughter board but I guess that using better fitting hardware is the better choice.