Co/Ni multilayers with a stack of Ta (2 nm)/[Co /Ni ] /Ta (30 nm) were prepared by dc magnetron sputtering, and their magnetization dynamics were measured by time-resolved magneto-optical Kerr effect (TRMOKE). The total thickness of the multilayer and perpendicular anisotropy were varied by changing the bilayer period and number of repeats while was kept at a constant of 2.5. The TRMOKE measurements show clear damped oscillation of the magnetization of Co/Ni multilayers after the pump pulse illumination, and the damping constant of the Co/Ni multilayers was estimated from the TRMOKE waveform. The estimated was found to be independent both on total thickness and anisotropy field of the multilayer and was estimated to be for all the multilayers. This means that the use of Ta capping and buffer layers is effective to evaluate intrinsic damping constant of the Co/Ni multilayer, and that independent control of and perpendicular anisotropy are possible for the magnetic multilayers.