We investigate the atomic structure and electronic properties of monolayers of copper phthalocyanines (CuPc) deposited on epitaxial graphene substrate. We focus in particular on hexadecafluorophthalocyanine (F(16)CuPc), using both theoretical and experimental (scanning tunneling microscopy - STM) studies. For the individual CuPc and F(16)CuPc molecules, we calculated the electronic and optical properties using density functional theory (DFT) and time-dependent DFT and found a red-shift in the absorption peaks of F(16)CuPc relative to those of CuPc. In F(16)CuPc, the electronic wavefunctions are more polarized toward the electronegative fluorine atoms and away from the Cu atom at the center of the molecule. When adsorbed on graphene, the molecules lie flat and form closely packed patterns: F(16)CuPc forms a hexagonal pattern with two well-ordered alternating α and β stripes while CuPc arranges into a square lattice. The competition between molecule-substrate and intermolecular van der Waals interactions plays a crucial role in establishing the molecular patterns leading to tunable electron transfer from graphene to the molecules. This transfer is controlled by the layer thickness of, or the applied voltage on, epitaxial graphene resulting in selective F(16)CuPc adsorption, as observed in STM experiments. In addition, phthalocyanine adsorption modifies the electronic structure of the underlying graphene substrate introducing intensity smoothing in the range of 2-3 eV below the Dirac point (E(D)) and a small peak in the density of states at ∼0.4 eV above E(D).