Can I cut the back off a china cabinet to convert it into a bookcase?
The back of the cabinet plays a large role keeping the unit square. My suggestion is to forget about trying to re-work that cabinet. There a bunch of concerns:


Cutting off the back 4 to 5 inches would remove the rear shelf support brackets.
The internal shelves are glass and would need to be cut or replaced with other materials.
There would still need to be a back to provide for the structural integrity of the unit.
You would still have to add a means to attach the thing to the wall. 


In the end the amount of messing around with it highly likely outweighs any benefit of trying to use this cabinet for another purpose. Instead invest in something that is already a wall mount shelf unit.