Here is a too-serious answer to your question, along with answers to a couple questions I think you should be asking:

The category you're interested in, as noted by others, is the category of coalgebras / corings, which is emphatically *not* the opposite category of rings --- but we're going to see exactly what's different between the two in the nicest case. To start things off, here's a definition of a coalgebra so that we're all on the same page: an $R$-coalgebra is an $R$-module $A$ together with morphisms $\Delta: A \to A \otimes_R A$ and $\epsilon: A \to R$ satisfying

- Coassociativity: $(\Delta \otimes \operatorname{id}) \circ \Delta = (\operatorname{id} \otimes \Delta) \circ \Delta$,
- Counitality: $(\epsilon \otimes \operatorname{id}) \circ \Delta = (\operatorname{id} \otimes \epsilon) \circ \Delta = \operatorname{id}$ after identifying $R \otimes_R A$ and $A \otimes_R R$ with $A$.

Coalgebras are familiar objects in algebraic topology, and you've already found the biggest source of them. Suppose you have a cohomology theory $E$ whose coefficient ring is a (graded) field. For any space $X$, the constant map $X \to \mathrm{pt}$ induces a map in homology $E_* X \to E_*$ which serves as the counit for $E_* X$. Toward a comultiplication, we have maps $E_* X \xrightarrow{\Delta} E_* (X \times X) \leftarrow E_* X \otimes_{E_*} E_* X$, but asking for the right-hand map to have an inverse is the same as asking for a Kunneth isomorphism. This only happens under restrictions on $X$ or restrictions on $E$ --- such as when $E_*$ is a field, which is one reason we requested that.

Now, let's lay our cards on the table and just announce some dualities we see in front of us:

- There's your opposite category $\mathsf{Rings}^{op}$, which is dual to rings in the sense that $(\mathsf{Rings}^{op})^{op}$ is equal to $\mathsf{Rings}$.
- In the language of the comments below, coalgebras are Eckmann-Hilton duals of algebras. This is really a statement about how we produced the definition above: we took the definition for an algebra, and we flipped all the arrows around.
- There's also the notion of linear algebraic duals: $V^\vee = \operatorname{Hom}_k(V, k)$. To avoid some very serious technicalities, we'll want to work in the nicest, most familiar setting possible: modules of finite rank over a ground ring that's a field.

Now, we would like to compare these three ideas. There is another category of interest floating around: the category of $k$-algebras has an associated category of presheaves $\widehat{\mathsf{Algebras}_k}$ $=$ $\operatorname{Functors}(\mathsf{Algebras}_k, \mathsf{Sets})$ which receives a map $\mathsf{Algebras}_k^{op} \to \widehat{\mathsf{Algebras}_k}$ described by the left side of the $\operatorname{Hom}$-functor: $X \mapsto \operatorname{Hom}(X, -)$. This assignment, called the Yoneda embedding, is a functor into a cocomplete category which is an equivalence onto its image and whose image is codense --- these are consequences of the Yoneda lemma. That the target of the Yoneda embedding is cocomplete makes it a much nicer category to play around in, and so it's worth considering what this embedding's use is.

I claim there's a relation between the category of $k$-coalgebras and $\widehat{\mathsf{Algebras}_k}$. Again, to make linear algebra behave nicely, we need to encode finiteness restrictions into our setup, and to make that happen we'll turn to "$k$-formal schemes". The classical $\operatorname{Spec}$ construction in algebraic geometry also gives a contravariant functor off the category of $k$-algebras which is an equivalence onto its image. Rather than fussing with what a Zariski spectrum is, since we're just playing around with categories, I will instead take the Yoneda embedding to be my definition of $\operatorname{Spec}$ and the presheaf category to be something dimly, vaguely, sorta like the category of schemes. Representable presheaves (i.e., those in the image of $\operatorname{Spec}$) are called affine schemes. Plenty of constructions from algebraic geometry transfer almost without comment; for instance, defining $\mathbb{A}^1 = \operatorname{Spec} \mathbb{Z}[x]$, we recover the functor $\mathcal{O}(X) = \operatorname{Hom}_{\widehat{\mathsf{Algebras}_k}}(X, \mathbb{A}^1)$, which in the case of an affine $X = \operatorname{Spec}(A)$ gives $\mathcal{O}(\operatorname{Spec} A) \cong A$.

A scheme $X$ will be called finite if it is $\operatorname{Spec}$ of an algebra of finite dimension as a $k$-module. These, too, are in ample supply in algebraic topology. If $X$ is a compact pointed space, then the algebra $H^* X$ will be finite in the sense we need. Of course, algebraic topology gets done on more than compact spaces, so we need to broaden our perspective a little bit: we can ask instead that $X$ be compactly generated, so that if $X_\alpha$ denotes the collection of compact subsets of $X$ directed by inclusion, we then have $X = \operatorname{colim} X_\alpha$. In the case that $X$ is a CW-complex, it is sufficient to take $X_\alpha$ to be just the finite subcomplexes of $X$. We might then be interested in the scheme $X_E := \operatorname{colim} \operatorname{Spec} E^* X_\alpha$. Such a scheme which occurs as the colimit of a directed system of finite $k$-schemes is called a $k$-formal scheme.

So now we have a suitable notion of a 'finite' scheme that still captures all our interesting (and frequently large) cohomology rings. Here's the comparison I promised early on: the functor taking a formal scheme $\operatorname{colim} \operatorname{Spec} A_i$ to the $k$-coalgebra $\operatorname{colim} A_i^\vee$ is an equivalence of categories. There's almost nothing to say because the sea of definitions we've made take care of most everything, but there is one key lemma: given a $k$-coalgebra, you need to know that you can write it as the colimit of finite $k$-coalgebras. The exact lemma that gets used is: if $E$ is a finite dimensional subspace of a $k$-coalgebra $A$, then there exists a subcoalgebra $F \subseteq A$ which is finite as a $k$-vector space and which contains $E$ (i.e., $E$ can be finitely enlarged so that it becomes closed under comultiplication). If you push around elements a bit you'll see that that's the case (and that this *requires* working over a ground field). Then, straight after, here's the second big assertion: the assignments $X \mapsto E_* X$ and $X \mapsto X_E$ are equivalent under the above equivalence of categories.

Now, this construction is delicate, and the limitations are not meant to be taken lightly! You need a ring structure on the underlying (co)homology theory to even dream of having products, you need Kunneth isomorphisms to make sense of the coalgebra structure, you need a coefficient field to have good linear-algebraic duality, and there are still potential problems with supercommutativity that we haven't addressed. But, when all the stars align and God smiles on us, this is what the coalgebra structure on homology is supposed to mean: it's another presentation of the formal scheme associated to the (perhaps more familiar) ring structure on cohomology.

To counterbalance that caveat, that's not to say that this point of view is not **immensely** useful. Here are some sample applications:

- All of chromatic homotopy: $\mathbb{C}\mathrm{P}^\infty$ ($ = B\mathrm{U}(1) = \mathrm{Gr}_1$) carries the structure of an $H$-group, and there are a whole sea of cohomology theories $E$, called complex-orientable, for which $\mathbb{C}\mathrm{P}^\infty_E$ is (noncanonically) isomorphic to $\hat{\mathbb{A}}^1 = \operatorname{colim} \operatorname{Spec} E^*[x] / \langle x^n \rangle$, an object which behaves a lot like an infinitesimal one-dimensional Lie group. This "formal Lie group" carries an immense amount of information about the cohomology theory $E$, and the "space" of available formal Lie groups carries an immense amount of information about stable homotopy theory as a whole.
- There are a variety of partial theorems in the following spirit: if $E$ and $F$ are complex oriented cohomology theories and $F$ is represented by the spaces $F_k$ in the sense that $F^k X = [X, F_k]$, then $\bigoplus_k E_* F_k$ behaves like $\operatorname{Hom}(\mathbb{C}\mathrm{P}^\infty_E, \mathbb{C}\mathrm{P}^\infty_F)$. Goerss has shown this when $E = H\mathbb{F}_p$ and $F$ satisfies a certain condition on $F_* \Omega^2 S^3$ (which is also satisfied for a class of complex-oriented spectra called Landweber exact). (Addendum: Goerss spends a lot of time setting up a "super" version of Dieudonne modules, which is meant to address in part issues with supercommutativity ignored/avoided here.)
- Armed with an ample supply of Kunneth isomorphisms, Ravenel and Wilson, the progenitors of the idea above, computed these coalgebras in the cases where $E$ and $F$ ranged in $K(n)$ (Morava $K$-theory), $E(n)$ (Johnson-Wilson theories), $BP$ (Brown-Peterson theory), and $H\mathbb{Z}/p^j$ (singular theories / Eilenberg-Mac Lane spaces). For instance, one can define the free (supercommutative) algebra on a group object in the category of $k$-formal schemes, and it turns out that $H_*(K(\mathbb{Z}/p^j, q); \mathbb{F}_p)$ is the free alternating algebra in this sense on the formal group scheme $B\mathbb{Z}/p^j_{H\mathbb{F}_p}$. A similar statement can be made for $K(n)_* K(\mathbb{Z}/p^j, q)$.
- The above ideas have stable versions too: the homology of spectra $E_* F$ is to be thought of as the scheme of isomorphisms $\operatorname{Iso}(\mathbb{C}\mathrm{P}^\infty_E, \mathbb{C}\mathrm{P}^\infty_F)$, and so, for instance, the dual of the Steenrod algebra, an object of classical interest, can be thought of as $\mathcal{O}$ of the automorphisms of a particular formal Lie group $\hat{\mathbb{G}}_a$. Relatedly, the statement that the dual Steenrod algebra coacts on the homology coalgebras $H_* X$ is straightened out in the category of formal schemes by saying that $\operatorname{Aut} \hat{\mathbb{G}}_a$ acts on the formal scheme $X_{H\mathbb{F}_2}$. This has also received classical interest, though not in this language: the final part of Thom's thesis on calculations in the real bordism ring amount to showing that this $\operatorname{Aut} \hat{\mathbb{G}}_a$ action on the homology of the real bordism spectrum is free.
- Finally, this is an example of a broader phenomenon: often the schemes associated to rings have enlightening interpretations in moduli theoretic terms. A great many classical objects in stable homotopy theory lead double lives in this framework as moduli spaces, and the geometry of the moduli space frequently informs us on how the original topological objects behave.

Anyway, this is all to say that you should definitely care deeply about the coalgebra structure on homology, as it's one way to get into formal schemes, where everything is interesting and magical and great.

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