$A$ is a countable set if and only if exist a injective function $h: A \to \mathbb{N} $ $A$ is a countable set if and only if exist a injective function $h:  A \to \mathbb{N}  $
I have the second implication: we know that $\mathbb{N}$ is countable then exist a injective function such that $h: A \to \mathbb{N}$  then $A$ is countable. 
But for first implication I don't know what I can do. I try with inverse function but, with my definition of countable set ($A$ is countable set if exist a surjective function $f: \mathbb{N} \rightarrow A$) and with composition I can't have the correct answer. 
 A: Suppose $A \neq \varnothing$ is countable. So there is $f: \mathbb{N} \to A$ surjective (your definition). To each $a \in A$, $f^{-1}(\{a\}) \neq \varnothing$, let $h(a) = \min f^{-1} (\{a\}) \in \mathbb{N}$  (because $\mathbb{N}$ is well ordered). Define $h: A \to \mathbb{N}: a \mapsto h(a)$.
Note, $$(f\circ h)(a) = f(h(a))  = f(\min f^{-1} (\{a\})) = a $$, then $h$ is injective. 
A: Thadeu Henrique Costa gave a succinct response, but for educational purposes it seems worthwhile to provide more material and another angle to the method.
The student should study
Injection iff Left Inverse
and to round out the course, 
Surjection iff Right Inverse.
We can avoid this theory for the OP's question, using, for instructional purposes, a direct approach:
By the Well ordering principle, every nonempty subset of $\mathbb{N}$ has a least element. So there is a function
$\tag 1 \mathcal M: \mathcal{P}^{*}(\mathbb{N}) \to \mathbb{N}$
mapping any nonempty subset of $\mathbb{N}$ to the smallest number of that set.
Exercise 1: Let $J$ and $K$ be two nonempty subsets of $\mathbb{N}$. Then
$\tag 2 J \cap K = \emptyset \; \Rightarrow \; \mathcal M(J) \ne \mathcal M(K)$
Now we can use the surjective $f$ enumeration to define another function
$\tag 3 \bar {f}^{-1}: A \to \mathcal{P}^{*}(\mathbb{N})$
by
$\tag 4 a \mapsto \{n \in \mathbb{N} \; | \; f(n) = a \} \ne \emptyset$
Exercise 2: Show that $\bar {f}^{-1}$ maps two different elements in $A$ to two  disjoint subsets of $\mathbb{N}$. In particular, $\bar {f}^{-1}$ is an injective function.
We know that the composition of two injective functions in an injective function. Now $\mathcal M$ is not an injective function, but by both exercise 1 & 2, it is injective on the image set $\bar {f}^{-1}(A)$ in $\mathcal{P}^{*}(\mathbb{N})$.
The answer to the OP's question is that $\mathcal M \circ \bar {f}^{-1}$ is an injective function from $A$ to $\mathbb{N}$. The OP should be able to easily check that this function is exactly the same as the function $h$ defined by Thadeu Henrique Costa.
