The usual argument (that you had trouble with) is not so complicated: you prove that every vector bundle is the pullback of the universal'' vector bundle on a Grassmannian. Then you compute the Chern classes of the universal bundle, directly on the Grassmannian, and by a miracle they turn out to be integral. In fact, you can compute the relevant integrals explicitly by hand, so it turns out quite nicely. So the hard bit is showing that every vector bundle is the pullback of the universal bundle; see Milnor and Stasheff, p. 65. As for Atiyah-Singer, that seems much more complicated and I don't see how to make use of it here.
The usual argument (that you had trouble with) is not so complicated: you prove that every vector bundle is the pullback of the universal'' vector bundle on a Grassmannian. Then you compute the Chern classes of the universal bundle, directly on the Grassmannian, and by a miracle they turn out to be integral. In fact, you can compute the relevant integrals explicitly by hand, so it turns out quite nicely. So the hard bit is showing that every vector bundle is the pullback of the universal bundle; see Milnor and Stasheff, p. 65. As for Atiyah-Singer, that seems much more complicated and I don't see how to make of it here.