I am mining with an 6870 so this is very well possible and that was my first rig also, so I went through the same learning curve.
The PC I used was running XP, so I had to download AMD Catalyst version 12.1 which includes Open C - this is required by the miner SW.
ALso make sure to use CGminer version 3.7.2 which is the last version that supports scypt (used for mining CGA)
Then you can check if the miner work on you computer by registering with a "mining pool" such as TheDigitalMint:
https://cga.thedigitalmint.net/index.php click on "Getting started" on the lower left and follow the directions.
You will then be able to start the miner and you will need to experiment with the settings for your card to get the best results.
Hint: look for no hardware errors (HW=0) and maximum count for Work Units (WU/minute) besides getting a good "hash rate".
What I am using for my old PC and the 6870 is:
--scrypt
--shaders 1120
-I 11 (highest w/o HW errors)
-w 64
-g 2 (default)
--thread-concurrency 2048
Note that all these settings as well as any over-clocking of the GPU core and memory are very dependent on the card,
so you will have to optimize them towards your card, but you can start with fewer setting as indicated in the "Getting Started"
directions on the pool and when it starts working, add more settings to optimize.
Success!
