A chaotic neural network reservoir (CNNR), in which a chaotic neural network model is used as a reservoir layer, efficiently introduces chaotic dynamics without violating the echo state property criterion. However, it is unclear which aspect of the complex reservoir dynamics contributes to the performance of CNNR. In this study, we analyze the dynamics of CNNR, which is used to engage time-series prediction tasks, and focus particularly on the design and evaluation of CNNR hardware. First, the memory capacity of CNNR is estimated using the NARMA-30 time series. Second, we analyze the dynamics of CNNR using hardware compatible indices such as the Lyapunov spectrum, permutation entropy, spatial mutual information, average firing rate, and Kullback-Leibler divergence. We show that any one of these indices alone is not enough to reveal the dynamics in CNNR. Therefore, we combined some of these indices to evaluate the dynamics further. The relationship between the prediction performance of CNNR and combined indices is identified. We finally introduce a complexity entropy causality plane to decipher the dynamics of CNNR in detail. As a result, we provide useful qualitative methods to evaluate and design the dynamics of CNNR hardware for time-series prediction tasks.