You will not have to repurchase anything. There is no process other than to install on the new machine using your disc and its serial number. You can download the installation files if need be as well.
You are allowed to have the software installed and activated on two machines, so if your old machine is on its way out but is still accessible you should deactivate that installation (open Acrobat and choose Help -> Deactivate) to free up the extra activation. If the old machine is not accessible you can contact Adobe support and they can perform the deactivation.
If the old machine is still usable you could also keep it activated there as a backup.