We study the graphite intercalated compound CaC₆ by means of Eliashberg theory. We perform an analysis of the electron-phonon coupling and define a minimal six-band anisotropic structure, which leads to a Fermi-surface dependence of the superconducting gap. A comparison of the superconducting gap structure obtained using the Eliashberg and the superconducting density functional theory is performed. We further report the anisotropic properties of the electronic spectral function, the polaronic quasiparticle branches, and their interplay with Bogoljubov excitations.