I'll chime in here since I have taken my cluster out a few times and restored most of the gauges.
You are going to have to take you switch panel out, the switch's below the instrument cluster. But before you un-screw and pull reach behind the dash and un plug the connectors to the switches. Being careful, it is possible to pull the switch panel forward some to see the light, dimmer, washer, wiper switches and the connectors.
Once out you will see all the screws holding in the cluster.
Again reach behind the cluster and carefully disconnect the speedometer cable and the round electrical connector going to the circuit board.
Now you can pull the cluster straight out towards you and take it out. in the last picture of the link tpodwdog sent you you see a few screws holding the circuit board in place along with the lamp sockets and a square tin box plugged into the circuit board. Replace that it is called a voltage limiter it takes the 12vdc from your car and converts it to 5vdc for the guages.
I say replace it because once you cluster is out you might as well update it to the new solid state version.
Now check all the screws holding the circuit board down and make sure they are tight. Re install the cluster and make sure every screw goes back in and is tight with good contact to the metal frame. Everything should be working perfectly. Good luck. Don't be afraid to tackle this fix we are here and can help you! post pictures as you go we can be like surgeons walking you through it from a remote camera
Remember this GROUNDS! everything in your car has to have a good ground for it to work electrically if you have one loose ground it will cause a problem. A loose screw at the cluster to dash frame a loose screw on the cluster circuit board. Even a painted surface can cause a bad ground.
I had a fried circuit trace and several loose pins on the round connector simple fix.
http://www.forbbodiesonly.com/moparforum/attachment.php?attachmentid=63311&stc=1&d=1334414004
Here are some simple wiring diagrams and below a more detailed technical one out of the service manual.
http://mymopar.com/downloads/1969/69BelvedereGTXSatelliteRoadRunnerA.JPG
http://mymopar.com/downloads/1969/69BelvedereGTXSatelliteRoadRunnerB.JPG
https://docs.google.com/open?id=0B20HDyEEpzYtZ1JVWHozUkhRMzZjV2s2QmsyekJSQQ