Perhaps not in his "Abelian varieties" book, but certainly in his "Tata lectures on Theta" Mumford describes this setup. See Chapter II of book 1. Another reference is Birkenhake and Lange's book "Complex abelian varieties" (see Chapters 3 and 8, for instance). By the way, I don't think you generally construct just $g$ independent meromorphic functions; you construct a whole bunch more of them - enough to embed the abelian variety into projective space.