Phylogenetics is a powerful tool for understanding the diversification dynamics of viral pathogens. Here we present an extension of the spectral density profile of the modified graph Laplacian, which facilitates the characterization of within-host molecular evolution of viruses and the direct comparison of diversification dynamics between hosts. This approach is non-parametric and therefore fast and model-free. We used simulations of within-host evolutionary scenarios to evaluate the efficiency of our approach and to demonstrate the significance of interpreting a viral phylogeny by its spectral density profile in terms of diversification dynamics. The key features that are captured by the profile are positive selection on the viral gene (or genome), temporal changes in substitution rates, mutational fitness, and time between sampling. Using sequences from individuals infected with HIV-1, we showed the utility of this approach for characterizing within-host diversification dynamics, for comparing dynamics between hosts, and for charting disease progression in infected individuals sampled over multiple years. We furthermore propose a heuristic test for assessing founder heterogeneity, which allows us to classify infections with single and multiple HIV-1 founder viruses. This non-parametric approach can be a valuable complement to existing parametric approaches.