All references below (unless otherwise stated) refer to Weibel (We): An Introduction to Hom. Algebra. Futhermore I consider projective resolutions and assume $\mathcal{A}$ has enough projectives and $F$ is right exact since this case is treated in Weibel. The case of injective resolutions can be easily adapted by switching to the opposite category [We, 2.3.4].
Let $Ch(\mathcal{A})$ be the category of (unbounded) chain complexes in $\mathcal{A}$. Since $\mathcal{A}$ is abelian, $Ch(\mathcal{A})$ is abelian as well [We, Th. 1.2.3] and has enough projectives [We, 2.2.2]. The functor $F: \mathcal{A} \to \mathcal{B}$ induces a functor
$Ch(F): Ch(\mathcal{A}) \to Ch(\mathcal{B})$. A morphism $h: C \to D$ of chain complees in $\mathcal{A}$ is epi, iff each $h_i:C_i \to D_i$ is epi [We, Proof of 1.2.3]. Hence, the right exactness of $F$ implies that $Ch(F)$ is also right exact.
In summary, we have shown: $Ch(F)$ is a right exact functor between abelian categories and $Ch(\mathcal{A})$ has enough projectives. Consequently, $Ch(F)$ has a left derived functor and everything what can be done for $L_\ast F$ (i.e. filtations, spectral sequences, etc.) can also be done for $L_\ast Ch(F)$.
A discussion of projective resolutions in $Ch(\mathcal{A})$ can be found in my answer to this question:
On the difference between a projective chain complex and a level-wise projective chain complex
Similar, a chain complex $I$ in $\mathcal{A}$ (which is now supposed to have enough injectives) is an injective object in $Ch(\mathcal{A})$, iff $I$ consists of injective objects $I_i\in \mathcal{A}$ such that all short sequences
$$ 0 \to \ker(d_i) \to I_i \to \operatorname{im}(d_i) \to 0$$
are exact and do split.
