Alternatively, if we take the definition of NP as the class of problems that can be verified in polynomial time, we can see that Careful 5-COL is in NP because there is a deterministic Turing machine that can check an answer in polynomial time.
In particular, given an instance $G$, we can take a certificate that $G$ is carefully 5-colorable as a function $f:V(G)\rightarrow \{0,\ldots,4\}$. Given this function, we can take each edge $uv$, and check that $f(v)-f(u) \equiv 1 (5)$. We only have to look at each edge once, so the algorithm to check is $O(|E|)$ (even if we're nasty about how $f$ is encoded, it's still something like $O(|E|\times|V|)$).