I always did the loopback test to see how far the signal was getting, so that I knew where the problem was. It's easy to do. Short the RX and TX pins on your USB adapter. Open a com program and connect, and type something random. If the serial device sends back exactly what you type, then it works. If it doesn't, your problem is with your serial adapter or driver. Get that to work first.
Next, you can short some RX/TX pins on your Arduino, to know if the signals are getting to the processor. I know how to do that on my Ardweeny, but you may have to do a web search how to do it on your Due.
On my Ardweeny, doing a loopback test would mean shorting the RX and TX pins (D0 and D1)
Also, a multi meter is very handy. Put one pin on ground, the other on the DTR pin. Get a com program that lets you toggle the DTR. I recommend Realterm. With loopback and multimeter, you should be able to trace the path of the signal and see where it stops. Best of luck!