What is the complexity class of finding vertex cover number of a simple graph?
It is actually finding the number that is NP-complete.  The (decision) problem is formally stated as follows:
Vertex cover.  Given a graph $G = (V, E)$ and an integer $k$, does there exist a set of at most $k$ vertices $C \subseteq V$ such that every edge has at least one endpoint in $C$?
As you can see, this is precisely your $\tau(G)$.
The other problem is often called the optimization problem, and is defined as following:
Vertex cover optimization.  Given a graph $G = (V, E)$, find a set $C \subseteq V$ of minimum size such that every edge has at least one endpoint in $C$.

The two problems are quite easy to jump between, at least if you are given polynomial time.  If you have an algorithm for the optimization problem, I'm sure you have already found out how to answer the decision version.
But given an algorithm for the decision version it is quite easy to create an optimization algorithm.  You can try to work it out yourself, but a hint is to see what happens with $\tau(G)$ if you remove a vertex from the graph.