I teach at a primarily undergraduate 4-year college in the US and we don't cover the proof of Cayley-Hamilton theorem in our linear algebra courses. I did however see both the computational and the abstract proof of the theorem when I studied from Hoffman-Kunze in my undergraduate (long time ago). I am wondering if anyone can share at what point in their institution's linear algebra courses the theorem is usually taught and at what level. Is it after eigenvalues/eigenvectors? Is it right after determinants are introduced? Is it taught in its most generality, such as "T is a linear operator over a vector field V with scalars k" or is it taught in a specialized case with linear transformations over C^n? I'm mostly interested to hear from people teaching at 4-year institutions in the US, but if the topic is taught at a 2-year college linear algebra course in the US, I would like to hear that too. Thank you for any information.
I see no compelling reason to teach Cayley-Hamilton in a first course directed at students who are not studying math as such. For math majors at a typical institution it probably makes more sense to teach Cayley-Hamilton in a first course on commutative algebra, rather than in a first-course on linear algebra. The principal use of Cayley-Hamilton in such courses is probably in the context of relating the minimal and characteristic polynomials and the deduction of canonical forms other than the Jordan form, and none of that material is typically included in the standard introductory linear algebra course. The theorem is not easy to state carefully (so that it does not appear to be a tautology) and the abstraction required to prove it is above the level of such a course. If one feels the need to indicate it, one can state it, and give exercises/examples checking it for particular linear transformations/matrices.
For what it's worth, I teach in Spain, in an engineering school. Our linear algebra class is taught in the first-year and has a higher level (more formal, more rigor) than what is standard in the sort of of US institution you ask about, but we do not teach Cayley-Hamilton.