In this paper, the implementation of a spectral or characteristic-based fifth order WENO (Weighted Essentially Non-Oscillatory) scheme is described in detail along with an adaptive multiresolution technique for efficiently computing the numerical solution of a multi-class traffic flow model described mathematically by a nonlinear system of conservation laws. In [1] the authors considered the same problem but in a component-wise manner. More recently, [2] conducted numerical experiments by using characteristic-based schemes combined with AMR (Adaptive Mesh Refinement) strategy.