Ni/NiO multilayers were grown by magnetron sputtering at room temperature, with the aid of the natural oxidation procedure. That is, at the end of the deposition of each single Ni layer, air is let to flow into the vacuum chamber through a leak valve. Then, a very thin NiO layer ($1.2 nm) is formed. Simulated x-ray reflectivity patterns reveal that layering is excellent for individual Ni-layer thickness larger than 2.5 nm, which is attributed to the intercalation of amorphous NiO between the polycrystalline Ni layers. The magnetization of the films, measured at temperatures 5-300 K, has almost bulk-like value, whereas the films exhibit a trend to perpendicular magnetic anisotropy (PMA) with an unusual significant positive interface anisotropy contribution, which presents a weak temperature dependence. The power-law behavior of the multilayers indicates a non-negligible contribution of higher order anisotropies in the uniaxial anisotropy. Bloch-law fittings for the temperature dependence of the magnetization in the spin-wave regime show that the magnetization in the multilayers decreases faster as a function of temperature than the one of bulk Ni. Finally, when the individual Ni-layer thickness decreases below 2 nm, the multilayer stacking vanishes, resulting in a dramatic decrease of the interface magnetic anisotropy and consequently in a decrease of the perpendicular magnetic anisotropy. V