I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant functor, i.e, $R^0 T$ is the reflection of $T$ in the category of functors that preserves cokernels.
Thanks in advance.
On
One can do this using satellites (see nlab) inductively. The classic reference (for module categories, but the same holds in general) is Homological algebra by Cartan-Eilenberg.
The modern way about thinking about derived functors stems from work of Quillen and Verdier in the 1960s and has a very pleasing formulation, albeit at the cost of introducing some fairly sophisticated machinery. I will try to explain this briefly.
Let $\mathcal{A}$ be an abelian category and let $\mathbf{Ch}^{\ge 0}(\mathcal{A})$ to be the category of cochain complexes (concentrated in degrees $\ge 0$). We define a quasi-isomorphism in $\mathbf{Ch}^{\ge 0}(\mathcal{A})$ to be a morphism that induces isomorphisms in cohomology. The derived category $\mathbf{D}^{\ge 0}(\mathcal{A})$ is the localisation of $\mathbf{Ch}^{\ge 0}(\mathcal{A})$ with respect to the quasi-isomorphisms, i.e. the category obtained by freely inverting all quasi-isomorphisms in $\mathbf{Ch}^{\ge 0}(\mathcal{A})$. Under good conditions, the category $\mathbf{D}^{\ge 0}(\mathcal{A})$ has a reasonably explicit description. For instance, if $\mathcal{A}$ is small, then its morphisms are zigzags of morphisms in $\mathbf{Ch}^{\ge 0}(\mathcal{A})$ modulo a certain equivalence relation; or if $\mathcal{A}$ has enough injectives, then there is a Quillen model structure on $\mathbf{Ch}^{\ge 0}(\mathcal{A})$ in which the weak equivalences are precisely the quasi-isomorphisms, and $\mathbf{D}^{\ge 0}(\mathcal{A})$ turns out to be equivalent to the category of degreewise injective cochain complexes with morphisms modulo homotopy.
Now, let $\mathcal{B}$ be an abelian category and let $T : \mathcal{A} \to \mathcal{B}$ be an additive functor. There is an obvious induced functor $T : \mathbf{Ch}^{\ge 0}(\mathcal{A}) \to \mathbf{Ch}^{\ge 0}(\mathcal{B})$, but it need not preserve quasi-isomorphisms. The Quillen–Verdier conception of derived functors, in retrospect, can be identified with certain Kan extensions and should be understood as the "best approximation" of arbitrary functors by functors that preserve quasi-isomorphisms. More precisely, let $\gamma_\mathcal{A} : \mathbf{Ch}^{\ge 0}(\mathcal{A}) \to \mathbf{D}^{\ge 0}(\mathcal{A})$ and $\gamma_\mathcal{B} : \mathbf{Ch}^{\ge 0}(\mathcal{B}) \to \mathbf{D}^{\ge 0}(\mathcal{B})$ be the localising functors. A total right derived functor for $T : \mathbf{Ch}^{\ge 0}(\mathcal{A}) \to \mathbf{Ch}^{\ge 0}(\mathcal{B})$ is then a left Kan extension of $\gamma_\mathcal{B} T$ along $\gamma_\mathcal{A}$, i.e. a functor $\mathbf{R} T : \mathbf{D}^{\ge 0}(\mathcal{A}) \to \mathbf{D}^{\ge 0}(\mathcal{B})$ equipped with a natural transformation $\eta : \gamma_\mathcal{B} T \Rightarrow (\mathbf{R} T) \gamma_\mathcal{A}$ such that, for any other functor $F : \mathbf{D}^{\ge 0}(\mathcal{A}) \to \mathbf{D}^{\ge 0}(\mathcal{B})$ and natural transformation $\alpha : \gamma_\mathcal{B} T \Rightarrow F \gamma_\mathcal{A}$, there is a unique natural transformation $\tilde{\alpha} : \mathbf{R} T \Rightarrow F$ such that $\tilde{\alpha} \gamma_\mathcal{A} \bullet \eta = \alpha$.
This must seem rather abstract and far-removed from old-style derived functors. To see the connection, we must find a way to compute total right derived functors. For this, it is useful to apply the technology of Dwyer, Hirschhorn, Kan, and Smith. We define a right deformation for $T$ to be a functor $R : \mathbf{Ch}^{\ge 0}(\mathcal{A}) \to \mathbf{Ch}^{\ge 0}(\mathcal{A})$ equipped with a natural quasi-isomorphism $i : \mathrm{id}_{\mathbf{Ch}^{\ge 0}(\mathcal{A})} \Rightarrow R$ such that, for $\mathcal{A}'$ the full subcategory of $\mathbf{Ch}^{\ge 0}(\mathcal{A})$ spanned by the image of $R$, the restriction $T : \mathcal{A}' \to \mathbf{Ch}^{\ge 0}(\mathcal{B})$ preserves quasi-isomorphisms. For instance, if $\mathcal{A}$ is a sufficiently nice abelian category, $R$ could be a functor that sends a cochain complex to a quasi-isomorphic cochain complex of injective objects. (Note that $T : \mathbf{Ch}^{\ge 0}(\mathcal{A}) \to \mathbf{Ch}^{\ge 0}(\mathcal{B})$ always preserves cochain homotopy equivalences, and any quasi-isomorphism of complexes of injectives is a cochain homotopy equivalence.) Regardless, suppose $(R, i)$ is a right deformation for $T$. It is clear that $T R : \mathbf{Ch}^{\ge 0}(\mathcal{A}) \to \mathbf{Ch}^{\ge 0}(\mathcal{B})$ preserves quasi-isomorphisms, so the universal property of $\mathbf{D}^{\ge 0}(\mathcal{A})$ implies there is a unique functor $\mathbf{R} T : \mathbf{D}^{\ge 0}(\mathcal{A}) \to \mathbf{D}^{\ge 0}(\mathcal{B})$ such that $(\mathbf{R} T) \gamma_\mathcal{A} = \gamma_\mathcal{B} T R$. Let $\eta : \gamma_\mathcal{B} T \Rightarrow (\mathbf{R} T) \gamma_\mathcal{A}$ be the natural transformation $\gamma_\mathcal{B} T i$.
Theorem. With notation as above, $(\mathbf{R} T, \eta)$ is an absolute left Kan extension of $\gamma_\mathcal{B} T$ along $\gamma_\mathcal{A}$. In particular, it is a total right derived functor for $T$.
The proof is not hard: it is just a matter of manipulating natural transformations. Hopefully, it is now clear that (when $\mathcal{A}$ is nice enough) the old-style $(R^n T) A$ is just the $n$-th cohomology of the cochain complex $(\mathbf{R} T) A$, where we are regarding objects in $\mathcal{A}$ as cochain complexes concentrated in degree $0$. (This makes sense because cohomology is invariant under quasi-isomorphism by definition!)