Natural transformations arise from arrow categories? We have a notion of morphisms-between-morphisms in category theory as the morphisms in an arrow category $\mathcal{C}^\to$. If we take $\mathcal{C}$ to be $\mathcal{Cat}$, the category of categories, then we obtain a ``natural'' notion of morphisms-between-functors!
How close is the relation of this notion to that of natural transformations? Is there some form of equivalence?
 A: Let $\mathbf{2}$ be the category with two objects and one arrow between them.  Then $\mathcal{C}^{\mathbf{2}}$ is what you are calling $\mathcal{C}^{\to}$.  Now, a functor $\mathcal{C}\to\mathcal{D}^{\mathbf{2}}$ is, via the cartesian closed structure of $\mathcal{Cat}$ (at least for the category of small categories), equivalent to $\mathbf{2}\to\mathcal{D}^\mathcal{C}$.  In other words, a functor into an arrow category is the same as an arrow in a functor category or a natural transformation. 
However, what you actually talk about is $\mathcal{Cat}^{\mathbf{2}}$ which has as objects functors like $F : \mathcal{C}\to\mathcal{D}$ and $G : \mathcal{E}\to\mathcal{F}$ and the arrows (from $F$ to $G$ say) are pairs of functors $H : \mathcal{C}\to\mathcal{E}$ and $K : \mathcal{D}\to\mathcal{F}$ such that $G \circ H = K \circ F$.  In other words, the arrows of an arrow category in general are just commutative squares of arrows of the category, and for $\mathcal{Cat}^\mathbf{2}$ are just commutative squares of functors.  These are definitely not natural transformations even when restricted to endomorphisms.
Clearly natural transformations, like any functor, embed as both objects and as arrows $Id\to Id$.  So $Nat(\mathcal{C},\mathcal{D})$ is $Hom(Id,Id)$ for $Id$ at say $\mathbf{2}\times\mathcal{C}\to\mathcal{D}$. Natural transformations are part of the 2-categorical structure of $\mathcal{Cat}$ and strict functors from $\mathbf{2}$ ignore that. Generalizing terminology, $\mathcal{C}^\mathbf{2}$ is the category of 1-cells in a bicategory $\mathcal{C}$.  Like any category, it's 1-cells are represented by $\mathbf{2}\to\mathcal{C}^\mathbf{2}$ or $\mathbf{2}\times\mathbf{2}\to\mathcal{C}$.  The 1-cells of $\mathcal{C}^\mathbf{2}$ are "1-cells between 1-cells" which need have no connection to 2-cells.  $\mathcal{Cat}$ is relatively special in being able to reify its 2-cells as 1-cells.  Looking at the 1-category of 1-cells and 1-cells between them in $\mathcal{Cat}$ is going to tell you something about the 2-cells only via this reification.  Since this reification doesn't exist in arbitrary bi- or 2-categories, you aren't going to find a pretty, "natural" description.
If you consider laxer notions of functor though, you can find more connections in basically the same way $\mathcal{C}^\mathbf{2}$ works, i.e. we can consider 2-functors from a 2-category containing one 0-cell, one 1-cell, and one non-identity 2-cell.
