Dictionary Learning (DL) is one of the leading sparsity promoting techniques in the context of image classification, where the "dictionary" matrix D of images and the sparse matrix X are determined so as to represent a redundant image dataset. The resulting constrained optimization problem is nonconvex and non-smooth, providing several computational challenges for its solution. To preserve multidimensional data features, various tensor DL formulations have been introduced, adding to the problem complexity. We develop a new alternating algorithm for the solution of the DL problem both in the matrix and tensor frameworks; in the latter case a new formulation based on Tensor-Train decompositions is also proposed. The new method belongs to the Proximal Alternating Linearized Minimization (PALM) algorithmic family, with the inclusion of second order information to enhance efficiency. We discuss a rigorous convergence analysis, and report on the new method performance on the image classification of several benchmark datasets.