You will need a serial port on your computer. If it doesn't have one you will need a USB to Serial (RS-232) converter.
Since the Copernicus uses 3.3V TTL logic you will need a RS-232 driver chip that can handle 3.3V logic. I've got one working with a MAX232 chip (which is actually 5V logic - use this to get around that issue: https://www.sparkfun.com/tutorials/65).
Then join all that together and it should work!