The separability of the massive Dirac equation in a rotating Kerr black hole background in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr spacetime is described in the framework of the Newman-Penrose formalism by a local Carter tetrad, and the Dirac wave functions are given on a spin bundle in a chiral Newman-Penrose dyad representation. Applying mode analysis techniques, the Dirac equation is separated into coupled systems of radial and angular ordinary differential equations. Asymptotic radial solutions at infinity and the event and Cauchy horizons are explicitly derived and, by means of error estimates, the decay properties are analyzed. Solutions of the angular ordinary differential equations matching the Chandrasekhar-Page equation are discussed. These solutions are used in order to study the scattering of Dirac waves by the gravitational field of a Kerr black hole. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in a Kerr background in horizon-penetrating coordinates, for the spectral theory of the corresponding Dirac Hamiltonian, and for the construction of an integral representation of the Dirac propagator.