Thin films of L1(0)-ordered (Fe51Pt49)(100-x)Cu-x with (001) texture were obtained by rapid thermal annealing of FePt(5 nm-t)/Cu(t) bilayers deposited by magnetron sputtering at room temperature on thermally oxidized Si(001) substrates. The L1(0) long-range ordering fully described by two independent chemical order parameters was investigated by anomalous x-ray diffraction measurements performed at the Fe K and Pt L-III edges. These measurements reveal that Cu atoms are preferentially located on the Fe-rich (001) planes. Up to a Cu content of 9 at.%, the Cu addition favors L1(0) ordering particularly after annealing to 600 degrees C, giving rise to an increase in effective magnetic anisotropy. Beyond this composition, Cu atoms displace a large fraction of Fe atoms to the Pt-rich (001) planes, leading to a rapid decrease in magnetic anisotropy and saturation magnetization.