Given a $\mathbb{k}$-algebra $A$ and two $A$-modules $M$ and $N$ find $\dim_{\mathbb{k}}\left(\mathrm{Hom}_{A}(M,N)\right)$ Let $\mathbb{K}$ be a field and $A$ a finite dimensional $\mathbb{k}$-algebra with identity $1_{A}$. For two $A$-modules $M$ and $N$ we have the set of all $A$-linear maps from $M$ to $N$ denoted $\mathrm{Hom}_{A}(M,N)$. This set is not in general an $A$-module neither by the left nor by the right (unless the $\mathbb{k}$-algebra $A$ is a commutative). 
Recall that any $A$-module $M$ is also a $\mathbb{K}$-vector space by the action:
$$\lambda\cdot m = (\lambda 1_{A}) m,\quad \forall\,m\in M\,,\lambda\in\mathbb{k}$$

In particular, the set of $A$-linear maps $\mathrm{Hom}_{A}(M,N)$ is also a $\mathbb{k}$-vector space using:

$$(\lambda\cdot f)(m)=(\lambda 1_{A})f(m).$$
It's known that  $\ \dim_{\mathbb{K}}(M)<\infty\ $  and $\ \dim_{\mathbb{K}}(N)<\infty\quad$ implies 
$$\quad \dim_{\mathbb{K}}\left(\mathrm{Hom}_{A}(M,N)\right)<\infty.$$

Is there any formula for
  $$\dim_{\mathbb{K}}\left(\mathrm{Hom}_{A}(M,N)\right)$$
  when $M$ and $N$ are finite dimensional $A$-modules, that is, $\dim_{\mathbb{K}}(M)<\infty$ and $\dim_{\mathbb{K}}(N)<\infty$ ?

we already know when $A=\mathbb{K}$ that:
$$\dim_{\mathbb{K}}\left(\mathrm{Hom}_{\mathbb{K}}(M,N)\right)=\dim_{\mathbb{K}}(M)\dim_{\mathbb{k}}(N)$$
 A: I don't know if that's what you're expecting, but there can be no formula in terms of $\dim_k(M), \dim_k(N)$ in the general case.
Case 1: For instance, if $A= k[T]/(T^2)$, $M$ is a $k$-vector space with a trivial action of $T$ and $N = A$, then for $f\in\hom_A(M,N)$, $x\in M$ one must have $Tf(x) = f(Tx)$, hence $Tf(x) = 0$ and so $f(x) =\lambda_x T$ for some $\lambda_x\in k$, and clearly $x\mapsto \lambda_x$ is $k$-linear. Conversely, if $l:M\to k$ is $k$-linear, then $x\mapsto l(x)T$ is $A$-linear, so $\dim(\hom_A(M,N)) = \dim\hom_k(M,k) = \dim(M)$
Case 2: However if $N$ is a $2$-dimensional $k$-vector space with a trivial action of $T$, then an $A$-module morphism $M\to N$ is just a $k$-linear map, so $\dim\hom_A(M,N) = \dim\hom_k(M,N) = (\dim M)(\dim N) =2(\dim M)$.
This shows that in general, there is no $F_A:\mathbb{N}^2\to \mathbb{N}$ such that $F_A(\dim M,\dim N) = \dim\hom_A(M,N)$
Case 3: There is an interesting case where there is a formula : when $A$ is a field and $A/k$ is thus a (finite dimensional) field extension. Then $\dim_A$ makes sense and the following formula holds for any $A$-module $M$ ($A$-vector space): $\dim_kM = (\dim_kA)(\dim_AM)$. Thus $\dim_k\hom_A(M,N) = (\dim_kA)(\dim_A\hom_A(M,N)) = (\dim_kA)(\dim_AM)(\dim_AN)= \frac{(\dim_kM)(\dim_kN)}{\dim_kA}$
