HI John,
You already have a system where you can read the 8051 serial stream on the PC's serial port, right?
Then, you need nothing more to use the HC-06. Just connect the HC-06 Tx and Rx to the 8051 Rx and Tx respectively (make sure you have the level shifters right, the HC-06 would require 3.3 volt swings), and you should be able to catch the data on the PC using any USB bluetooth dongle running on a virtual com port. You can confirm the 2-way comms using a terminal emulator on the PC.
Regards,
Anand Dhuru