Hello!
A typical approach to this image would be to separate the jaw from the head:
Create a bone to control the head, and a bone to control the jaw, then turn the jaw into a mesh and bind it to both bones:
the parts that should be connected to the head get 100% head bone influence, everything else can get the jaw influence.
You can see a similar example of jaw but at a different angle in the dragon project that you can download here:
Spine: Twitch
At 1h 10s I start rigging the jaw just like explained above:
The different angle may just require you to translate the jaw bone instead of just rotating it, and if you're after a 2.5D effect as you seemed to mention, you can even constraint that bone to the bone that controls the 2.5D effect setting the translation to a number inverse or that has a lower influence that 100.
