Why minimum vertex cover problem is in NP
You don't have to verify that $V'$ is minimum.
The decision version of Vertex Cover (which you have quoted in your question) only asks you to decide whether there is a vertex cover of size at most $k$.
To verify that $V'$ is a valid yes-certificate for an instance $\langle G=(V,E), k \rangle$ you just check that:

$V' \subseteq V$,
$|V'| \le k$, and
$\forall (u,v) \in E, \{u,v\} \cap V' \neq \emptyset$.