Man-made artificial graphene has attracted significant attention in the past few years due to the possibilities to construct designer Dirac fermions with unexpected topological properties and applications in nanoelectronics. Here we use a first-principles approach within density-functional theory to study molecular graphene similarly to the experiment by Gomes et al. [Nature (London) 483, 306 (2012)NATUAS0028-083610.1038/nature10941]. The system comprises carbon monoxide molecules arranged on a copper (111) surface in such a way that a honeycomb lattice is obtained with the characteristic electronic properties of graphene. Our results show in detail how carbon monoxide molecules modify the copper surface (and regions beneath) and create a honeycomb lattice of accumulated electrons between the adsorbate molecules. We also demonstrate how the properties of the formed Dirac fermions change as the CO density is tuned, and provide a direct comparison with experimental scanning tunneling microscope images.