We have deposited epitaxial Ni/Cu bilayers and Cu/Ni/Cu trilayers on single-crystal (100) Si substrates in a molecular beam epitaxy chamber. The magnetic properties were studied in situ using magneto-optic Kerr effect measurements, and ex situ using vibrating sample magnetometry. The preferred direction of magnetization is perpendicular to the film plane across a broad region of Ni thickness. The thickness dependence of the anisotropy energy has been explained using the Neel pair interaction model and in terms of carefully characterized strains which are thickness-dependent due to misfit relaxation. We have observed misfit dislocations in these films using transmission electron microscopy and measured their average separation as a function of Ni thickness. Additionally, strain in the Ni layers has been measured as a function of Ni thickness using substrate curvature measurements. Using these results, we have calculated the value of the surface magnetic anisotropy term, which we have shown to be strain dependent.