Phosphorene is an emerging 2D-like material with direct energy band. In this work we report the results of a theoretical study on the electronic structure of phosphorene multilayers. A particular emphasis is put on the investigation of the optical absorption and the functionalization of phosphorene layers with organic molecules such as benzene and fullerene. The investigation is carried out employing the density functional theory, and the effect of using different exchange-correlation functionals for the interlayer van der Waals interaction is discussed. Fundamental quantities like lattice constants, interlayer distance and energy band gap are reported in phosphorene monolayers, bilayers and trilayers. The features of the interband optical absorption are studied from the calculated imaginary part of the dielectric function. The results of the numerical simulation of the phenomenon of the small organic molecule physisorption onto phosphorene indicate that the direct band gap is preserved. In the case of the fullerene physisorption, a deformation in the phosphorene monolayer is induced, leading to a shift of the associated band structure. It is shown that such a modification depends on the particular exchange-correlation functional employed. In the case of benzene physisorption, the electronic structure of the phosphorene remains unchanged and is independent of the position of the benzene molecule. This suggests that benzene would be a good candidate for a molecular coating of phosphorene to shield it against oxidation under ambient conditions.