This paper deals with the problem of H-infinity control via measurement feedback with internal stability for discrete-time nonlinear systems. Using the theory of discrete-time dissipative systems and differential game, we first present a parameterization of a family of static state feedback H-infinity nonlinear controllers. Then, we provide sufficient conditions for the existence of a dynamic output compensator which solves the problem of disturbance attenuation with internal stability for discrete-time nonlinear systems. The theory developed in this paper incorporates the well-known results for discrete-time linear systems, and generalizes some recent advances in H-infinity control of continuous-time nonlinear systems to their discrete-time counterparts.