When we think of groups as one object categories, how do we define the morphisms for particular group elements?

If a group is thought of as a category with just one object, which we might denote *, then an element of the group becomes a morphism from * to itself (so is an automorphism of the object *).

The `category' $\mathbb{Z}_3$ has a single object. The three elements of $\mathbb{Z}_3$ are morphisms from * to *. 0 is the identity morphism on * and $1:*\to *$ is another morphism, and the composition of $1$ with itself gives us $2:*\to *$. NB: * is just an abstract object and is not $\mathbb{Z}_3$ as you tried to write. About the only thing you can know about this object is that it has exactly two endomorphisms other than the identity and they are both invertible, so they are abstract automorphisms. The automorphisms of * form a group isomorphic to $\mathbb{Z}_3$.

I think your final questions are based on a confusion, so they do not quite make sense.

The related question: Confused about the definition of a group as a groupoid with one object. may help.