**EDIT** Corrected a couple of inaccuracies and mistakes, added some references.

For the sake of clarity, let me work with non-negatively graded cochain complexes, and analyze the case of a left exact covariant functor, to prove that its right derived functor is a universal (covariant) cohomological $\delta$-functor.

Let $\mathcal A$ and $\mathcal B$ be two abelian categories, and let $F : \mathcal A \to \mathcal B$ be a left exact functor. Let me denote by $\gamma_\mathcal{A}$ the localization functor $\mathrm{Ch}_+(\mathcal{A}) \to \mathrm{D}_+(\mathcal{A})$. Let $\tilde S_\mathcal{A}$ denote the full subcategory of $\mathrm{Ch}_+(\mathcal A)$ consisting of discrete complexes (i.e. of complexes whose differentials are all zeroes). Let $S_\mathcal{A}$ denote its essential image in the derived category.

First of all, observe that a $\delta$-functor is a particular case of what MacLane calls "connected sequences" (connected sequence is a $\delta$-functor iff the long sequence it induces is exact). Now, defining a connected sequence is equivalent to define a functor $S_\mathcal{A} \to \mathcal{B}$ (see Proposition XII.8.1 and Proposition XII.8.2 in [1]). Part of the equivalence works as follows: given any $T : S_\mathcal{A} \to \mathcal{B}$, define $T^n(A) := T(A[n])$, and check that, given any exact sequence $0\to A\to B\to C\to 0$, the image of the class corresponding to it in
$$S_\mathcal{A}(C[n],A[n+1]) \simeq \mathrm{D}_+(\mathcal A) (C[n], A[n+1]) \simeq \mathrm{Ext}^1_{\mathcal A}(C,A)$$
under $T$ gives you a morphism $\delta^n : T^n C \to T^{n+1} A$ giving $\{T^n\}$ the structure of a connected sequence. You can find the details of this construction in [1].

We have that
$$\mathbb{R}F := \mathrm{Lan}_{\gamma_\mathcal{A}(-[0])} (\gamma_\mathcal{B} F(-)[0]).$$
Now, we can postcompose the bottom corners of the square defining $\mathbb{R}F$, along the functors $$\gamma_{\mathcal A} \circ \bigoplus_{n\geq 0} H^n : \mathrm{D}_+(\mathcal{A}) \to S_\mathcal{A}$$ and
$$H^0 : \mathrm{D}_+(\mathcal{B}) \to \mathcal{B}$$ as follows

$$
\require{AMScd}
\begin{CD}
\mathcal{A} @>{F}>> \mathcal{B}\\
@VVV @VVV \\
\mathrm{D}_+(\mathcal{A}) @>{\mathbb{R}F}>> \mathrm{D}_+(\mathcal{B})\\
@VVV @VVV \\
S_\mathcal{A}@. \mathcal{B}
\end{CD}
$$

and further Kan extend, obtaining the connected sequence $RF : S_\mathcal{A} \to \mathcal{B}$ as
$$\mathrm{Lan}_{\gamma_\mathcal{A} \circ \oplus_n H^n \circ (-)[0]}(H^0 \circ (-)[0] \circ F) \simeq \\ \simeq \mathrm{Lan}_{\gamma_\mathcal{A} (-[0])}(F).$$
Now, given any other connected sequence (in particular, any $\delta$-functor) $T : S_\mathcal{A} \to \mathcal{B}$, by definition of left Kan extension we have
$$\mathrm{Nat}(RF,T) \simeq \\ \simeq \mathrm{Nat}(F, T \circ \gamma_{\mathrm{A}} (-[0])) \simeq \\ \simeq \mathrm{Nat}(F,T^0)$$
showing that the "universality" of $RF$ works not only for $\delta$-functors, but in general for connected sequences.

The fact that for $F$ left exact and $\mathcal A$ with enough injectives one has that $RF$ is not only a connected sequence, but really a $\delta$-functor, follows from XII.8.3, 4 and 5 in [1].

[1] MacLane - Homology