I have my doubt about the next definition:
Let $M\subset R^n$ be a surface m-dimensional of class $C^k$. We say a function $f:M\longrightarrow R^l$ is class $C^r$ with $0<r<k+1$, if $f ( \phi )$ is class $C^r$, with $\phi :V_0\longrightarrow V$ parameterization $C^k$.
For surface I understand the next:
A surface m-dimensional and class $C^k$ on $R^n$, is a set $M\subset R^n$ such that for every $p\in M$ there exist a open set $U_\alpha \subset R^n$ such that $p\in U_\alpha \cap M=V_\alpha $, and a parameterization m-dimensional and class $C^k$, $\phi _\alpha: V_0\longrightarrow V_\alpha $.
My question is: why must $r$ be less than $k+1$?...has sense if $r>k$?...for instance, May I have a function $f$ on a surface $C^k$, being $f$ class $C^{k+1}$?
To define the regularity of a function $f\colon M\to \mathbb R$ you need to compose it with a parameterization, so you define $$ f_\alpha:=f\circ \phi_\alpha\colon V_0\to \mathbb R.$$ Here I am adopting your notation, and in particular, I remark that $V_0$ is an open set in $\mathbb R^m$ (you did not state this precisely - please pay attention to this, it is very important).
We say that $f\in C^\ell(M)$ if and only if $M$ is covered by coordinate patches $V_{\alpha_1}, V_{\alpha_2},\ldots$ in such a way that $f_{\alpha_j}\in C^\ell(V_0)$. Notice that, since $V_0$ is an open set in Euclidean space, $C^\ell(V_0)$ is defined in terms of partial derivatives, as is standard in calculus.
It is desirable that this definition automatically implies that all $f_\beta$ are $C^\ell$, regardless of whether $\beta$ is one of the $\alpha_j$ or not. Now, for all $\alpha$ and $\beta$, we have the change of parameterization formula $$ f_\beta=f_\alpha\circ(\phi_\alpha^{-1}\circ\phi_\beta).$$ Note that this function is defined on the open set $\phi^{-1}(V_\alpha\cap V_\beta)$, provided this is not empty.
We notice that the transition function $\phi_\alpha^{-1}\circ\phi_\beta$ needs not be more regular than $C^k$, where $k$ is the regularity of the manifold $M$. Therefore, even if $f_\alpha$ is smoother than $C^k$, after composing with the transition function it might lose some regularity, and we can only guarantee that $f_\beta$ is $C^k$.
We conclude that the highest possible level of regularity that a function $f\colon M\to \mathbb R$ can achieve is $k$.