The formation of carbonaceous deposits on Pt(111) surfaces and Pt nanoparticles has been studied using suitable models and density-functional calculations. The study addresses a broad range of processes, from the very first stage of carbon deposition up to a final building of graphene monolayers (ML) defined as a 1:1 ratio of the number of C atoms to surface Pt atoms. A carbidic phase is formed below a coverage of similar to 0.3 ML, when negatively charged carbon atoms are strongly adsorbed preferentially on fcc hollow sites. On Pt nanoparticles, the adsorption of carbon atoms seems to be enhanced near particle edges due to the special flexibility of defect sites. Above a coverage of similar to 0.3 ML, the formation of small C-n aggregates becomes possible. Interestingly, thermodynamics favors the formation of C-3 trimers at a coverage of 0.33 ML, whereas the formation Of C-2 dimers requires a higher coverage of 0.5 ML. The covalently bonded C2 species is supposed to be the key fragment for the formation of benzene-like rings at coverages above 0.6. ML. These rings are expected to be the building blocks for the graphene monolayer. However, the typical electronic structure of graphene is not observed until a coverage above similar to 1.8 ML is reached. We corroborated the experimentally suggested carbon double-layer to be stable. It is proposed to consist of a monolayer of carbidic atoms C adsorbed on Pt with a graphene layer adsorbed on the carbidic layer. Some of the carbidic atoms serve as anchors for the graphene layer, with noticeably strong covalent bonds formed. This double-layer model would imply a much higher adhesion of the graphene layer than in the single-layer model.