In this paper, we consider a smoothing kernel-based classification rule and propose an algorithm for optimizing the performance of the rule by learning the bandwidth of the smoothing kernel along with a data-dependent distance metric. The data-dependent distance metric is obtained by learning a function that embeds an arbitrary metric space into a Euclidean space while minimizing an upper bound on the resubstitution estimate of the error probability of the kernel classification rule. By restricting this embedding function to a reproducing kernel Hilbert space, we reduce the problem to solving a semidefinite program and show the resulting kernel classification rule to be a variation of the k-nearest neighbor rule. We compare the performance of the kernel rule (using the learned data-dependent distance metric) to state-of-the-art distance metric learning algorithms (designed for k-nearest neighbor classification) on some benchmark datasets. The results show that the proposed rule has either better or as good classification accuracy as the other metric learning algorithms.