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Single unit-cell FeSe films grown on (001) SrTiO₃ have attracted extensive theoretical and experimental attention because they have a significantly enhanced superconducting critical temperature (T_c) compared to that observed in bulk FeSe crystals. This is due in part to finely tuned levels of electron doping at the interface. Here we describe the growth of Co_xFe_1-xSe thin films by molecular beam epitaxy on (001) SrTiO₃. We investigate the superconducting properties of these films over the complete concentration range (0 < x < 1) and study the evolution of their electronic structure using laser-based angle resolved photoemission spectroscopy. By increasing the Co concentration, we demonstrate a suppression in the superconducting critical temperature compared to pure FeSe films, along with the emergence of a concomitant, and possibly competing, ferromagnetic order with perpendicular anisotropy. Additionally, we inspect the distribution of substituted Co atoms using scanning transmission electron microscopy and energy-dispersive x-ray spectroscopy.