We present first-principles calculations of the electronic band structure and spin-orbit effects in graphene functionalized with methyl molecules in dense and dilute limits. The dense limit is represented by a 2 x 2 graphene supercell functionalized with one methyl admolecule. The calculated spin-orbit splittings are up to 0.6 meV. The dilute limit is deduced by investigating a large, 7 x 7, supercell with one methyl admolecule. The electronic band structure of this supercell is fitted to a symmetry-derived effective Hamiltonian, allowing us to extract specific hopping parameters including intrinsic, Rashba, and pseudospin inversion asymmetry spin-orbit terms. These proximity-induced spin-orbit parameters have magnitudes of about 1 meV, giant compared to pristine graphene whose intrinsic spin-orbit coupling is about 10 mu eV. We find that the origin of this giant local enhancement is the sp(3) corrugation and the breaking of local pseudospin inversion symmetry, as in the case of hydrogen adatoms. Similarly to hydrogen, also methyl acts as a resonant scatterer, with a narrow resonance peak near the charge neutrality point. We also calculate STM-like images showing the local charge densities at different energies around methyl on graphene.