Let's say we have a (not transitive) model, $(M,E)$, of Z. In this post, "class" means an arbitrary collection of elements of $M$. Let $x_0\ni x_1\ni x_2\ni\cdots$ be an infinite descending chain of elements of $M$. Then $X=\{x_i:i\in\mathbb{N}\}$ is countable (outside of $M$), but $X$ cannot be a set in $M$ since it violates Foundation. Hence, $M$ sees $X$ as a proper class.
Anything wrong with this reasoning?
Well, your use of "class" here is a bit odd. Usually a class is required to be definable in some sense, and $X$ is definitely not. Indeed, re: your last sentence I would say that $M$ doesn't "see" $X$ in any sense whatsoever.
However, if you simply use "proper class in the sense of $M$" to refer to any subset of $M$ which is not in $M$ (or, more accurately, is not of the form $\{b\in M: b\in^Ma\}$ for some $a\in M$), then yes, $X$ is a proper class in the sense of $M$.