Why are homomorphisms of groups the best choice of morphisms in category of groups? Well, "groups" in the question could be replaced by orther algebraic structures.
For example, why don't we let the morphisms be maps between groups not preserving group structure? 
The reason I think of is that if we do this, when defining object by universal property, we just get objects up to bijection, rather than up to isomorphism of groups, which is not the way we want them behave.
But are there some other reasons? 
 A: Let's say that you have two groups: $G$ and $H$, for example
$$G=\{0,1,2,3\}$$
$$H=\{5,6,7,8\}$$
I didn't define what group operations are on both $G,H$ but is that necessary? Yes, it is, because we know that there are two groups of order $4$ up to isomorphism. But functions that do not preserve group structure do not care about that. From function's point of view these two groups are the same.
But wait, up to what? Here's the problem. Let's say that $G=\{0,1,2\}$ and since it is of order $3$ then we know that there is a unique (up to isomorphism) group structure on $G$. But we can define a group structure on $G$ in multiple ways (for example by having $0$, $1$ or $2$ as the neutral element) but in reality they are all the same because we can freely "translate" the group structure between them. It's a matter of looking at $G$ from a proper angle.
So what is that "translation"? Nothing else then an invertible homomorphism, i.e. an isomorphism. Necessity is the mother of invention.
So the reason we use homomorphisms is simply because it is a tool that allows us to analyze relationships between groups. Functions that do not preserve group structure don't really tell us much about that relationship.
Now we can obviously analyze the category of groups with simple functions as morphism. But we don't do that because that doesn't lead to anything useful. Simple as that.
Final note: it doesn't mean that the classical homorphism is the best choice. It only means that it is a better choice then plain functions and we didn't find a better candidate yet. But note that sometimes new candidates for morphisms do appear almost like out of nowhere, e.g. the case of the homotopy category.
