None of the factors you list about Charlie committing a crime, about to commit a crime* or being responsible for the officer's injury apply to whether or not Bob can shoot Charlie on behalf of the officer.
On the other hand, if Charlie's actions lead Bob to the reasonable belief that Charlie represents an imminent threat of death or grave bodily injury to anyone then Bob will be justified in using force, including deadly force, to stop Charlie. I'm aware of many states that also allow the use of deadly force to prevent the imminent commission of a forcible felony.
A normal citizen can go to great lengths to assist the police and stories of citizens coming to the aid of police officers are easily found. There may be some reason the police officer felt the need to use deadly force against Charlie. Bob will need to make his own determination based on all the factors available to him, which may include communication from the officer. If Bob decides to shoot Charlie then an investigation will center on whether it was reasonable for Bob to deduce that the use of deadly force was justified given the information available to Bob at the time.
Some states have laws that penalize refusal to aid a police officer. A list of them can be found on this wikipedia page. Notably, California just repealed their statute requiring assistance.
*The statutes of some states allow the use of deadly force to prevent the commission of felonies, typically forcible felonies. However, just because a crime is listed as a forcible felony doesn't always mean that deadly force can be used to stop the crime. An example comes from Florida where the statutes state that deadly force can be used to stop forcible felonies, the statutes also define burglary as a forcible felony. However, the courts have ruled that the use of such force is not reasonable where the structure being burglarized is unoccupied.