The capacity-achieving input covariance matrices for coherent block-fading correlated MIMO Rician channels are determined. In this case, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. An approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to $+\infty$. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information, while being much more computationally attractive.