Expression (2) of the OP is the definition of the effective action $\Gamma^{1}_{\text{eff}}$, which depends on $A_\mu$,
$$\text{exp} \left (i \, \Gamma^{1}_{\text{eff}}[A_\mu]\right)\equiv \int \mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left( i \int \bar{\Psi} (i {\not} D-m) \Psi \right). \tag 2$$
This allows to write the generating function, eq. (1) of the OP, as
$$
Z[j_\mu]=\int \mathcal{D} A_\mu \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu}+i\, \Gamma^{1}_{\text{eff}}[A_\mu]+\int j_\mu A_\mu\right), \tag 1$$
where I have added explicitly a source $j_\mu$. Note that we have gained nothing with writing the generating function that way, since it is still as hard to compute exactly than before.
However, it might be a good starting point to perform some approximation. Since the functional integral in (2) is gaussian, we can formally compute it and obtain (up to some signs, factor $i$, etc.)
$$
\Gamma^{1}_{\text{eff}}[A_\mu]={\rm Str} \log (i\not D-m), \tag 3
$$
which can be expanded in power of $A_\mu$. To lowest orders, one obtain a renormalization of the propagator (due to a vacuum polarization bubble), then a $4$-photon interaction and so on, i.e.
$$
\Gamma^{1}_{\text{eff}}[A_\mu]={\rm cst}+\frac12\int \Gamma^{1,(2)}_{{\rm eff},\mu\nu} A_\mu A_\nu+\frac1{4!}\int \Gamma^{1,(4)}_{{\rm eff},\mu\nu\lambda\sigma} A_\mu A_\nu A_\lambda A_\sigma+\cdots,
$$
where $\Gamma^{1,(n)}_{{\rm eff}}$ is the $n$-th functional derivative with respect to $A$ (evaluated in $A=0$ or in a constant field $A^0$ depending on the problem at hand), and can be computed explicitly from (3). These vertex functions can be written in terms of one-loop diagram of fermions with $n$ inclusions of the bare fermion-photon vertex.
The higher loop correction you would get from the standard QED perturbative expansion will come from the loop-expansion of (1).
