It's my opinion that derived functors are better understood through a universal property. On derived categories it is possible to express the condition of derivability in a concise way due to the additivity of everything in sight. My impression is that you do not need the full machinery of homotopical algebra unless you are in a non-additive situation (which is often the case, I should add).

Let me spell it out. Let $F\colon A\to B$ be a left exact functor between Abelian categories. Being additive, this induces a (Delta) functor between their *homotopy* categories that by abuse of notation we keep calling it $F\colon K(A)\to K(B)$. Recall that $K(A)$ is the category of (possibly unbounded) complexes of objects of $A$ with maps homotopy classes of chain maps. Now if we localize a homotopy category making invertible the quasi-isomorphisms (i.e. those maps that induce isomorphisms in homology) we obtain the derived category $D(A)$. If we call the canonical localization functor $Q \colon K(A) \to D(A)$ and similarly for $B$, the problem of deriving $F$ is to extend "in an optimal way" $Q \circ F$ to $D(A)$. The solution, if it exists, it is expressed by the following universal property:

The (Delta) functor $RF \colon D(A)\to D(B)$ is the derived functor of $F$ if there is a natural transformation $Q \circ F \to RF \circ Q$ such that for any other (Delta) functor $G \colon D(A)\to D(B)$ the natural transformation induces a bijection
$$ Hom(RF, G) \longrightarrow Hom(Q \circ F, G \circ Q)$$

I suggest you draw a diagram (It is difficult to draw a diagram here, but very instructive if you do).

It turns out that if there are unbounded injective resolutions on $K(A)$ they provide a right adjoint to $Q \colon K(A) \to D(A)$ that automatically gives a way to express $RF$. This is related to the Bousfield localization philosophy in the context of derived categories.

Yet another question is the description of the adjoint functor i.e. the unbounded injective resolution. If the complex is bounded below, then the resolution can be constructed step by step. In the case that $A$ is the category of modules over a ring, one can use the exactness of products and use countable homotopy limits as in Bökstedt-Neeman. In more general cases such as Grothendieck categories, one needs more sophisticated tools.