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The electronic structure of thin films of FeTe grown on Bi$_2$Te$_3$ is investigated using angle-resolved photoemission spectroscopy, scanning tunneling microscopy and first principles calculations. As a comparison, data from cleaved bulk \FeTe taken under the same experimental conditions is also presented. Due to the substrate and thin film symmetry, FeTe thin films grow on Bi$_2$Te$_3$ in three domains, rotated by 0$^{\circ}$, 120$^{\circ}$, and 240$^{\circ}$. This results in a superposition of photoemission intensity from the domains, complicating the analysis. However, by combining bulk and thin film data, it is possible to partly disentangle the contributions from three domains. We find a close similarity between thin film and bulk electronic structure and an overall good agreement with first principles calculations, assuming a p-doping shift of 65~meV for the bulk and a renormalization factor of around 2. By tracking the change of substrate electronic structure upon film growth, we find indications of an electron transfer from the FeTe film to the substrate. No significant change of the film's electronic structure or doping is observed when alkali atoms are dosed onto the surface. This is ascribed to the film's high density of states at the Fermi energy. This behavior is also supported by the ab-initio calculations.