We use an atomistic approach to study the electronic properties of monolayer and bilayer black phosphorus in the vicinity of a charged defect. In particular, we combine screened defect potentials obtained from first-principles linear response theory with large-scale tight-binding simulations to calculate the wave functions and energies of bound acceptor and donor states. As a consequence of the anisotropic band structure, the defect states in these systems form distorted hydrogenic orbitals with a different ordering from that in isotropic materials. For the monolayer, we study the dependence of the binding energies of charged adsorbates on the defect height and the dielectric constant of a substrate in an experimental setup. We also compare our results with an anisotropic effective mass model and find quantitative and qualitative differences when the charged defect is close to the black phosphorus or when the screening from the substrate is weak. For the bilayer, we compare results for charged adsorbates and charged intercalants and find that intercalants induce more prominent secondary peaks in the local density of states because they interact strongly with electronic states on both layers. These insights can be directly tested in scanning tunneling spectroscopy measurements and enable a detailed understanding of the role of Coulomb impurities in electronic devices.