We present a fully relativistic numerical method for the study of cosmological problems using the Baumgarte-Shapiro-Shibata-Nakamura formalism on a dynamical Friedmann-Lemaître-Robertson-Walker background. This has many potential applications, including the study of the growth of structures beyond the linear regime. We present one such application by reproducing the Lemaître-Tolman-Bondi solution for the collapse of pressureless matter with arbitrary lapse function. The regular and smooth numerical solution at the center of coordinates proceeds in a natural way by relying on the partially implicit Runge-Kutta algorithm described in Montero and Cordero-Carrión [arXiv:1211.5930]. We generalize the usual radiative outer boundary condition to the case of a dynamical background and show the stability and convergence properties of the method in the study of pure gauge dynamics on a de Sitter background.