Let $U$ be a countable collection of subsets of $X$. Then there exists a countable set which consists of elements of the elements in $U$.
Proof: As $U$ is a countable collection of subsets of $X$, it follows that the elements in $U$ can be indexed by the natural numbers. Furthermore, by the axiom of choice we can choose an element from each element of $U$ and form a new set $C$ which consists of elements of each element in $U$. So $C = \{ x_n : x_n \in U_n\}$ so we can define a map $f$ from $C$ to $U$ by $f(x_n)=U_n$ then it suffices to show that the map is injective, thus making $C$ countable. Suppose $f(x_n)=f(x_m) = U_n =U_m$: as $U$ is countable, $n=m$ hence $x_n=x_m$ so the map is injective and therefore $C$ is countable.
Is the proof correct? How could I improve it?
Your proof is mostly correct. It however needs correction on these two points:
$\boxed{\textit{Firstly:}}$ the axiom of choice is used on collections of non-empty sets and you never forbade your collection $U$ from containing $\emptyset$ as that is still a valid subset of $X$.
Hence you want to eliminate the empty set from your collection first before using the axiom of choice on it. That is you want proceed in your proof with $U' = U - \{\emptyset\}$ instead or say something to the effect of "without loss of generality, assume that $U$ does not contain $\emptyset$; otherwise just replace $U$ with $U - \{\emptyset\}$".
$\boxed{\textit{Secondly:}}$ this map you are defining from $C$ to $U$:
may not be well-defined. For instance, if $x_{1,2} \in C$ were a common element of $U_1$ and $U_2$, then do you define $f(x_{1,2})$ as $U_1$ or $U_2$? And your hypotheses never mentioned that $U$ is a collection of disjoint subsets of $X$. Hence it is totally possible for $U_1$ and $U_2$ to have non-empty intersection.
The correct way to go about it is this. Simply define a map from $C$ to $\Bbb N$ directly like so (there is no need to map into $U$):
For each $x \in C$, certainly $x \in U_k$ for some $k \in \Bbb N$. So the set $C_x \subseteq \Bbb N$ of all $n \in \Bbb N$ such that $x \in U_n$ is non-empty (as $n = k$ is in there). Thus the well-ordering of $\Bbb N$ says that the least element of $C_x$ must exist; so we can define $f(x) = \min C_x \in \Bbb N$.
You can check that this gives an injective map from $C$ into $\Bbb N$ due to the uniqueness of minimums. And you can now conclude that $C$ is countable (where I am assuming finiteness is a possibility when I say "countable").