How to fish thermostat cable using an existing cable
That's a very labor intensive approach considering some of the latest technologies. Have you thought about converting to a Wireless Thermostat? 
Take a look at the Honeywell YTH6320R1001. 



You might also consider a cheaper option with the VentStar Add-A-Wire. This setup allows you to use your existing four wires and still be able to have a common wire. It's an ingenious idea that piggybacks unused wires and converts them into one common wire.