We develop a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states (ABS) in interacting quantum dots connected to superconducting leads. A local effective Hamiltonian for dressed ABS, including both the atomic (or molecular) levels and the induced proximity effect on the dot is argued to be a natural starting point. A self-consistent expansion in single-particle tunneling events is shown to provide accurate results even in regimes where the superconducting gap is smaller than the atomic energies, as demonstrated by a comparison to recent numerical renormalization group calculations. This simple formulation may have bearings for interpreting Andreev spectroscopic experiments in superconducting devices, such as scanning tunnel microscope measurements on carbon nanotubes or radiative emission in optical quantum dots.