It seems to me that everything written so far is addressing the title, not the body of the question, in particular focusing on the unstable. Part of that is people asking their own questions. But I think it is best to start with the easiest questions. Indeed, Tom, you have singled out the right place to start: Yes, the Atiyah-Hirzebruch spectral sequence is a hypercohomology spectral sequence.
Of course, we need a generalization or abstraction of the hypercohomology spectral sequence to make sense of that. Surely, you know that as you move from homological algebra to stable homotopy theory, you should generalize from derived categories of abelian categories to the abstraction of triangulated categories.* But there is another abstraction that you should know about, that of t-structures.
As Dylan says, spectral sequences come from filtrations.** But where do filtrations come from? One source is taking a chain complex and filtering by degree. The subquotients reveal exactly the original chain complex, while we'd prefer something that depended only on the quasi-isomorphism class. (Though often this is good enough because, although its $E_1$ page is not canonical, the rest of the sequence is.) The steps of this filtration are called the "naive truncations." They have the property that their homology agrees with the original in low degrees, is zero in high degrees, and is not canonical in a single degree. With a little modification, this can be changed into the good truncation, which has no such intermediate degree, but goes directly from agreeing to disagreeing.
Here is one version of the hypercohomology spectral sequence, I think pretty close to way that you see it. Start with a right-derivable functor $F\colon A\to B$ between abelian categories and the goal of understanding its derived functor on chain complexes $RF\colon DA\to DB$. Start with a complex $C\in DA$, take its good filtration, so that its subquotients are $H^iC[i]$, shifts of objects of $A$, and apply $RF$ to get a chain complex made of $R^jF(H^iC)$. Then as you reassemble $C$ from the $H^iC$, the spectral sequence reassembles $RF(C)$ from $RF(H^iC)$.
Let us abstract what this argument required: not that $DA$ was the derived category of an abelian category, but merely that each object had a filtration whose subquotients lived in shifts of an abelian category, called the heart; and that it was easier to understand the functor restricted to this abelian category. The first condition, which does not mention the functor, is called a t-structure.
Ultimately, the point is that the stable homotopy category has a t-structure whose heart is the category of abelian groups, but exotic t-structures abound and the concept was introduced in algebraic geometry. The easiest example is given by duality: The derived category of perfect chain complexes of abelian groups (that is, bounded complexes of finitely generated free abelian groups) is equivalent to its opposite under the contravariant duality functor $\mathrm{Hom}(-,\mathbb Z)$. Thus, we can transport the t-structure across the duality to get a new t-structure on the old category. The old heart is the category of finitely generated abelian groups (placed in degree zero). The new heart is equivalent to the opposite of the old heart. It consists of objects that are the sum of a free abelian group in degree zero and a torsion group shifted by $1$.
The filtration from the t-structure on the stable homotopy category is called the Postnikov filtration. The objects of its heart are called the Eilenberg-MacLane spectra; their associated cohomology theories are usual cohomology. Thus if our functor of interest is cohomology of a fixed space $X$ with coefficients in the varying spectrum $E$, its restriction to the heart is ordinary cohomology, a well-understood starting point, and so the hypercohomology spectral sequence is $H^i(X;E^j)\Rightarrow E^{i+j}(X)$, the same form as the Atiyah-Hirzebruch spectral sequence. (You may have to do some work to check that it is the actually the same spectral sequence.)
* Probably what I have to say can be made to work under Verdier's axioms for triangulated categories, but I really mean stable $\infty$-categories. Or you could work with DG-categories, until you want to move to stable homotopy theory.
** The Bockstein spectral sequence is a purely algebraic spectral sequence that I do not know how to see as coming from a filtration, though I have a vague memory of another spectral sequence that does come from a filtration and carries the same information.
Best Answer
Chain complexes arise from simplicial abelian groups via the Dold–Kan correspondence: the normalized chain complex functor establishes an equivalence of categories from simplicial abelian groups to (nonnegatively graded) chain complexes.
Likewise, bicomplexes arise from simplicial objects in chain complexes (or bisimplicial abelian groups): the normalized chain bicomplex functor establishes an equivalence of categories from simplicial objects in chain complexes (or bisimplicial abelian groups) to bicomplexes.
Under this correspondence, the totalization simply computes (up to a quasi-isomorphism) the realization (alias homotopy colimit) of the corresponding simplicial object. This is precisely the content of the Dold–Puppe version of the Eilenberg–Zilber theorem.
The result continues to hold for unbounded chain complexes: the totalization of a chain complex of unbounded chain complexes computes the homotopy colimit of the corresponding simplicial object.
Simplicial objects are very common in homological algebra (e.g., the bar construction), so bicomplexes and totalizations fall out naturally.