I have some ideas about how to define morphism of morphisms in category theory, but I don't know anything about higher category theory. The idea is quite simple, and can be iterated easily. I'll only focus on Sets, but it is clear that the following concept could easily be categorifiable.
Let's give us some algebraic structure (magma, monoid, group, vector space, $\ldots$), and let $f : A \rightarrow B$ be a morphism of such structures. Notice that the graph of $f$, $\Gamma(f) = \{ (x, f(x)) \}$ inherits automatically of the same structure as $A$ and $B$, simply by defining the binary law as $(x,f(x)) * (y, f(y)) = (x*_A x', f(x) *_B f(y))$.
Let $f : A \rightarrow B$ and $g : C \rightarrow D$ two morphisms of the same type of structure. Call 2-morphism $\alpha : f \rightarrow g$ any morphism $\alpha : \Gamma(f) \rightarrow \Gamma(g)$, that is, any morphism of their graph. Clearly, $\Gamma(\alpha)$ also inherits of the same structure as $A$, $B$, $C$, and $D$, allowing us to iterate the process.
Is it known ? If not, does this sound a good idea to consider higher morphisms ?
As it turns out, $\Gamma(f)$ is simply isomorphic to $A$. Let $\varphi: A \to \Gamma(f)$ be defined by $\varphi(a) = (a, f(a))$. Then $$\varphi(a*b) = (a*b, f(a*b)) = (a,f(a)) * (b,f(b)) = \varphi(a) * \varphi(b).$$ And $\varphi$ has an obvious inverse, namely the projection on the first coordinate. Finally $A \cong \Gamma(f)$.
If you want to consider "morphisms of morphisms", there are various possibilities. One of them is the arrow category $\mathsf{Ar}(\mathsf{C})$ for example, where objects of $\mathsf{Ar}(\mathsf{C})$ are morphisms $f : A \to B$ in $\mathsf{C}$, and morphisms of $\mathsf{Ar}(\mathsf{C})$ are commutative diagrams. For example a morphism from $(f : A \to B)$ to $(g : C \to D)$ looks like: $$\require{AMScd} \begin{CD} A @>{f}>> B \\ @VVV @VVV \\ C @>{g}>> D \end{CD}$$
This isn't really related to higher category theory, though.