Biochemistry is what is says - the study of the chemistry of living things.
Since all living systems are complicated, chemical systems, biochemistry is the study of how living things work. Often biochemists are called cell biologists, molecular biologists or biotechnologists depending on where they are and what they do. I guess at one time one could write a distinct definition for each. Often persons with degrees in each will cooperate on the same project. In truth, the distinctions are unimportant.