This is (embeddable mouth rig v00) an easy rig to use because it has really 8 controls to move for the mouth and one control to move the jaw.
The first figure shows the general hierarchy of the rig.
The rig is basically made up of a main rig, which is fed by two local rigs. I first learned about using multiple local rigs from Jason Osipa's Stop Starring book (pg 322, third edition).
The second figure shows one way of connecting a local rigs control to the main rigs control, here I used expressions.