This tutorial is a work in progress. Remove this warning when it will be tested (and somehow indicate on which devices it was tested on).
Follow the instructions for setting up ADB on your computer so that you can access a root shell on your device.
NOTE: when prompted on your Replicant device, make sure that you check the box that says Always allow from this computer when you grant your computer USB debugging permissions. Otherwise, you will be unable to obtain root shell access on your Replicant device when you reboot it into the recovery OS to actually perform the backup.
NOTE: for security reasons, you may want to revoke these non-expiring permissions once the backup is complete.
To reboot in the recovery, you can follow the instructions in the RebootIntoTheRecovery wiki page.