Plenty of books dedicate the first parts of the course to the experimental evidence of the need of a new theory, namely:the ultraviolet catastrophe of the blackbody radiation, the photoelectric effect, the unexplained stability of the electron position in an atom among others. From there you introduce, the Planck constant, the quantum of light concept, the de Broglie particle-wave duality and the Schrodinger equation and off you go. It is a pretty standard approach. As per the study of QM as a mathematical probabilistic theory you dont need Newtons Laws at all. The 5,6 postulates are pretty standard, and you take the position operator and the Hamiltonian definitions as part of them , no need to relate them to classical counterparts, at least until you prove the Ehrenfest theorem.