Rapid thermal annealing was applied to transform sputter-deposited Fe51Pt49/Cu bilayers into L10 chemically ordered ternary (Fe51Pt49)100-xCux alloys with (001) texture on amorphous SiO2/Si substrates. It was found that for thin film samples, which were processed at 600 °C for 30 s, the addition of Cu strongly favors the L10 ordering and (001) texture formation. Furthermore, it could be revealed by transmission electron microscopy and electron backscatter diffraction that the observed reduction of the ordering temperature with Cu content is accompanied by an increased amount of nucleation sites forming L10 ordered grains. The change of the structural properties with Cu content and annealing temperature is closely related to the magnetic properties. While an annealing temperature of 800 °C induces strong perpendicular magnetic anisotropy (PMA) in binary Fe51Pt49 films, the addition of Cu systematically reduces the PMA. However, due to the enhancement of both the A1-L10 phase transformation and the development of the (001) texture with increasing Cu content, lowering of the annealing temperature leads to a shift of the maximum perpendicular magnetic anisotropy towards alloys with higher Cu content. Thus, for an annealing temperature of 600 °C, the highest perpendicular magnetic anisotropy energy is found for the (Fe51Pt49)91Cu9 alloy. The smooth surface morphology, adjustable PMA, and high degree of intergranular exchange coupling make these films suitable for post-processing required for specific applications such as for sensorics or magnetic data storage.