We propose a new first-principles-based procedure to determine diffusion coefficients in metals and dilute alloys. In particular, we compute the following kinetic quantities entirely from first-principles: the formation/migration enthalpies and entropies of vacancies and solute atoms.  The attempt frequency for diffusion is also calculated.  We illustrate the method by computing the self-diffusion coefficient of fcc Al and fcc Cu and the diffusion coefficients of Mg, Si and Cu in Al individually, through a vacancy mechanism. In the case of the dilute alloys we use the fcc-based five-frequency model to describe diffusion. We obtain results from both the local-density approximation (LDA) and the generalized-gradient approximation (GGA).  The proposed method yields self-diffusion and impurity diffusion coefficients that are in good agreement with existing experimental measurements.