When are kernels (resp. cokernels) finite limits (resp. finite colimits)? Under what conditions, in what contexts, are kernels (resp. cokernels) finite limits (resp. finite colimits)?
 A: I misunderstood the question slightly (I thought you were asking about equalizers and coequalizers). The most general definition of a kernel or cokernel that I'm aware of takes place in a category with zero morphisms. This includes categories with zero objects as well as $\text{Ab}$-enriched categories such as abelian categories.
In such a category, the kernel (resp. cokernel) of a map $f : a \to b$ may be defined as the equalizer (resp. coequalizer) of $f$ and the zero morphism $0 : a \to b$. In particular, kernels are a finite limit and cokernels are a finite colimit. In an $\text{Ab}$-enriched category, the equalizer of an arbitrary pair of parallel morphisms $f, g : a \to b$ is the kernel of $f - g$, so being able to compute kernels (resp. cokernels) is equivalent to being able to compute equalizers (resp. coequalizers). 
You can verify that this reproduces the usual notion of kernel and cokernel of modules, for example. It also gives a notion of kernel and cokernel for, say, pointed topological spaces. 
A more general definition replaces the notion of kernel and cokernel with the notion of kernel pair and cokernel pair. This is a generalization of the equivalence relation on a set $X$ induced by a map $f : X \to Y$ and is a pullback, so in particular a finite limit. Cokernel pairs are dually given by pushouts, so in particular are finite colimits. 
