This paper presents formulation and results of a self-consistent dynamical theory which can describe the crossover from the local-moment state to the heavy-fermion state with decreasing temperatures. The theory is an extension of the non-crossing approximation (NCA) for the single-site model and incorporates the intersite interaction via self-consistent modification of the conduction-electron propagator. Detailed numerical results for single-particle excitations are presented which show that (i) the cooperative enhancement of the Kondo resonance occurs at temperatures higher than the Kondo temperature T _K ; (ii) a pseudogap develops around the energy of the Kondo resonance with decreasing temperatures; (iii) no new energy scale other than T _K is generated in the heavy-fermion state.