I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite well, except for some inconsistence (from my point of view) in using left/right cosets for actions of the modular group on the set of matrices of determinant $n$. I overcome this trouble by using one more simple arithmetic lemma which relates left and right cosets. I wonder whether there exist another version of the Kronecker--Weber approach for modular polynomials, or maybe even another elementary proof.
I find Silverman's exposition in his "Advanced Topics in the Arithmetic of Elliptic Curves" very clear, and concise. Look at the section on the integrality of the j-invariant, in particular Theorem 6.1 of Chapter II, Section 6, pages 140-151. You can find three proofs of the integrality of the j-invariant and, more concretely, you should have a look at the "Analytic proof of Theorem 6.1", starting in page 143, which is the "classical" one I believe you are interested in (it is also a proof using matrices of determinant n, but the exposition is great).