We consider the hydrodynamic description of a fluid of particles in the context of the classical approach to the Nonequilibrium Statistical Operator Mehod. It is based on the information entropy ensemble of Predictive Statistical Mechanics, and its accompanying Informational Statistical Thermodynamics. We start with a description of the macroscopic state of the system in terms of single- and two-particle reduced dynamic density functions in phase space, and the accompanying Lagrange multipliers (intensive nonequilibrium thermodynamic variables) that the method introduces. In terms of this basic set of dynamical variables we derive the equations of evolution for the mass, momentum, and energy densities, as well a the continuity equation for the informational entropy. It is shown how these equations are to be restricted in order to recover the results of classical hydrodynamics (based on linear irreversible thermodynamics), as well as a Gibbs relation defining local equilibrium. The differences between the generalized formalism and this classical limiting case are discussed.