A fully relativistic approach is presented for the calculation of the bremsstrahlung emitted by an electron scattered off an ionic target. The ionic target is described as a combination of a nuclear electrostatic potential and a screening potential induced by the electronic cloud of the ion. The approach allows us to investigate the influence of the target electrons on the properties of the emitted radiation. We calculate the double differential cross section and Stokes parameters of the bremsstrahlung of an electron scattered off uranium ions in different charge states, ranging from bare to neutral uranium. Results on the high-energy endpoint of the electron bremsstrahlung from Li-like uranium ions U89+ are compared to the recent experimental data. For this process, it is found that taking into account the screening effect leads to a change of the cross section on the level of 14%, which can, in principle, be seen in present-day experiments.