Asking for a deformation over $\mathbb{A}^1$ is quite restrictive.  Even asking for formal deformations / deformations over an étale cover of $\mathbb{A}^1$ is nontrivial.  The "standard" obstruction group for deforming a stable map is the hyper-Ext group $\mathbf{R}Hom^2_{\mathcal{O}_{C_0}}(L^\bullet_{f_0},\mathcal{O}_{C_0})$, where $L^\bullet_{f_0}$ is the dualizing complex of $f$, i.e., it is (globally) quasi-isomorphic to the two-term complex,
$$ NL^{\bullet}_{f_0}: \ f_0^*\Omega_{X_0/k} \to \Omega_{C_0/k}, $$
concentrated in degrees $-1$ and $0$.  

In some form, this is described in Behrend-Fantechi, particularly the last few sections.  I also recommend the first chapter of Kollár's textbook, "Rational Curves on Algebraic Varieties".  Sernesi's book on Hilbert schemes is also great.  Debarre's book is wonderful, and very readable, ...