Pontryagin duality is the statement that there's a natural isomorphism between the identity functor on $\mathsf{LCA}$, the category of locally compact (Hausdorff) abelian groups with continuous homomorphisms as arrows, and the functor $\chi ^2$, where $\chi$ takes objects of $\mathsf{LCA}$ to their dual group and acts on arrows via precomposition.
Now, I see everywhere that the natural transformation $1\overset{\centerdot}{\rightarrow}\chi^2$ (soon to be proven a natural isomorphism) is specified component-wise in the following manner $$ [\rho _G(x)](\alpha)=\alpha (x)\;\;\;\;\;\;\;\;\;\; x \in G,\;\alpha\in \chi (G)$$ What bugs me is the dependence on $\alpha$. Each component $\rho_G$ is supposed to be a function $\rho_G:G\rightarrow \chi ^2 (G)$, so its only input should be $x\in G$ and I'd expect $\alpha$ to be generated by $x$ somehow, but this doesn't seem to be the case.
If I understand correctly, any assignment $x\mapsto \alpha \in \chi (G)$ generates a natural transformation and there's nothing canonical about which one we take at all. Anyway, now we take $\alpha \mapsto (\alpha \mapsto \alpha(x))$ and supposedly the entire process is natural, which makes sense because we have in a way "cancelled" the arbitrariness of the assignment $x\mapsto \alpha$ by then looking at $\alpha$ at this very $x$.
So how confused am I? Is the assignment $x\mapsto \alpha$ indeed arbitrary? If so, what is the formal role of $\alpha$ here, a parameter? Would any assignment $x\mapsto \alpha$ indeed induce a natural transformation?
There is no dependence on $α$. As you say, $ρ_G$ is a function $G → χ^2(G)$, so $ρ_G(x)$ is an element of $χ^2(G)$, and in particular a function $ρ_G(x) : χ(G) → \mathbb T$. For $α ∈ χ(G)$, it's value $[ρ_G(x)](α)$ is defined to be $α(x) ∈ \mathbb T$.
This is exactly analogous to the standard embedding of a vector space into its double dual, ie. to the natural monomorphism $V → V^{**}$, and on the level of underlying sets, it's just the map $X → Y^{Y^X}$, given by $x ↦ \mathrm{ev}_x$.