What kind of object is the kernel of a ring homomorphism? The category $\mathbf{Grp}$ of groups has a zero object, namely the trivial group $1$. Since $\mathbf{Grp}$ is furthermore complete, we have the notion of a kernel of a group homomorphism. The kernel of a homomorphism $G\xrightarrow{\ f\ }H$ is a monomorphism $\operatorname{Ker}f\xrightarrow{\ker f}G$, so $\operatorname{Ker}f$ can be interpreted as a subgroup of $G$ with underlying set a subset of $G$. Namely, $\operatorname{Ker}f=\{x\in G\mid fx=1\}$.
In the category $\mathbf{Ring}$ of rings, we don't have a zero object and thus there is no natural definition of a kernel. Anyway, we define the kernel of a ring homomorphism $R\xrightarrow{\ f\ }S$ as the set $\operatorname{Ker}f=\{x\in R\mid fx=0\}$, analogous to the case of (additive) groups. However, it is not clear to me, what kind of object this is, i.e. to which category it belongs.
It seems wrong to think of it as a set, because it does have extra structure, e.g. as an (abelian) group. So maybe one should define the kernel of a ring homomorphism as an abelian group (the kernel of the underlying group homomorphism). But also this seems a little arbitrary, because one can easily define Rings and kernels of such without ever coming across abelian groups. Also, this doesn't give rise to a one-to-one correspondence, since not every kernel of a homomorphism of underlying abelian groups gives rise to a kernel of a ring homomorphism.

So, is it possible to define a kernel of a ring homomorphism without "leaving" the category of rings?

The same problem comes with all ideals. Is an ideal a set? A group? An abelian group? A module? A non-unitary ring? If one indeed defines left-ideals of $R$ as submodules of $R$, viewed as a left-module over itself, a right-ideal as a left-ideal of $R^{\operatorname{op}}$ and a two-sided ideal as a left-and-right ideal, how does this intuitively give rise to the fact that
two-sided-ideal$\iff$kernel of underlying group homomorphism of some ring homomorphism?
I hope you can understand the trouble I have.
 A: Yes. Instead of taking the kernel you can take the kernel pair. In general, the kernel pair of a morphism is an attempt to recover a universal equivalence relation compatible with that morphism. If $f : R \to S$ is a morphism, then the kernel pair of $f$ is the pullback of the diagram $R \xrightarrow{f} S \xleftarrow{f} R$. In the case of rings, if $f$ has kernel $I$ then this is the ring
$$\{ (r_1, r_2) \in R \times R : r_1 \equiv r_2 \bmod I \}.$$
This sort of object is called a congruence in universal algebra, and should be thought of as an equivalence relation internal to the category of rings. One way to state one of the isomorphism theorems is that $f$ is surjective iff it is the coequalizer of the two projections from its kernel pair to $R$, or equivalently iff it is an effective epimorphism. 
For groups, abelian groups, and rings, the ability to take inverses (in the third case, for addition) turns out to imply that you can replace the study of kernel pairs with the study of kernels (in the third case, at the price of leaving the category, as you noticed). But when you can't do this you really need to look at kernel pairs; for example if you start studying monoids or semirings. 
See this blog post for a non-categorical introduction to the idea of internal equivalence relations and these three posts for variations on the relationship between kernel and cokernel pairs and various flavors of monomorphism and epimorphism. For example, with no hypotheses on the category, a morphism is a monomorphism iff its kernel pair exists and is trivial and, dually, an epimorphism iff its cokernel pair (the kernel pair in the opposite category) exists and is trivial. 
A: A left ideal of a ring $R$ is just a left $R$-submodule of $R$. Thus, the natural category in which left ideals live is the category of left modules. The latter has a zero object and kernels can be computed as usual. If $f : R \to S$ is a homomorphism of rings, you can view $S$ as an left $R$-module via $r s := f(r) s$, so that $f$ becomes a homomorphism of left $R$-modules and the kernel of this homomorphism (in the category of left $R$-modules) is what you usually call the kernel of the ring homomorphism of $f$. Similarly we can deal with right ideals. And for two-sided ideals we work with the category of $(R,R)$-bimodules and observe that every ring homomorphism $R \to S$ induces an $(R,R)$-bimodule structure on $S$. By the way, all these definitions also work when we replace $\mathsf{Ab}$ by any abelian tensor category. For example, (quasi-coherent) ideal sheaves also fit into this picture.
A: Viewing rings as the commutative algebra objects of a symmetric monoidal category seems to make the setting more amenable to abstraction (which seems to be the goal here). So do that: $\mathbf{CRing} = \mathbf{CAlg}(\mathbf{Ab})$, where $\mathbf{Ab}$ is the symmetric monoidal category of abelian groups.
Hovey defines a Smith ideal (in https://arxiv.org/abs/1401.2850) as a monoid (in the pushout product monoidal structure, see paper) in the arrow category $\mathrm{Arr}(\mathcal{C})$ for a monoidal category $(\mathcal{C},\otimes,\mathbb{1})$.
So a Smith ideal for $\mathbf{Ab}$ ends up being an ideal inclusion $(j : I \to R) \in \mathrm{Arr}(\mathcal{C})$  
