In this paper, we propose a new speech enhancement system using a deep neural network (DNN)-augmented colored-noise Kalman filter. In our system, both clean speech and noise are modelled as autoregressive (AR) processes, whose parameters comprise the linear prediction coefficients (LPCs) and the driving noise variances. The LPCs are obtained through training a multi-objective DNN that learns the mapping from the noisy acoustic features to the line spectrum frequencies (LSFs), while the driving noise variances are obtained by solving an optimization problem aiming to minimize the difference between the modelled and observed AR spectra of the noisy speech. The colored-noise Kalman filter with DNN estimated parameters is then applied to the noisy speech for denoising. Finally, a post-subtraction technique is adopted to further remove the residual noise in the Kalman-filtered speech. Extensive computer simulations show that the proposed speech enhancement system achieves significant performance gains when compared to conventional Kalman filter based algorithms as well as recent DNN-based methods under both seen and unseen noise conditions.