Surface ordering of pentacene molecules adsorbed on an aperiodic Cu surface has been studied with density functional theory (DFT) and scanning tunnelling microscopy as a function of coverage. Below 0.73 ML (5.3 × 1013 molecules cm-2), the adsorbate structure is row-like with the molecular axes aligned with the rows in the Cu structure. Between this coverage and 1 ML (7.3 × 1013 molecules cm-2), a structural phase with a checkerboard structure is seen. At this coverage region, the molecules are very close to each other which leads to unusual bending. At higher coverages, a further phase transition to a high-density row structure is seen for most of the film. DFT with van der Waals functionals is employed to study how the molecule-molecule and molecule-surface interactions evolve as a function of coverage.