When we think of groups as one object categories, how do we define the morphisms for particular group elements? I have been studying category theory for fun, and got confused on a concept. So apparently we can view a group as a one object category where the morphisms are the group elements
and we define the composition of maps by the product of the group elements. This all makes sense to me, till I get really technical. For example, a morphism is a function from one object to another. So let us look at $Z_{3}$, we have the group element 1. If we want to think of 1 as a morphism we have to technically define a map from $Z_{3}$ to $Z_{3}$. Since the group is cyclic I will define the map $1:Z_3 \rightarrow Z_3$ by $0\rightarrow 1$, $1\rightarrow 2$, and $2 \rightarrow 0$. Similarly, define the map $2:Z_{3}\rightarrow Z_{3}$ by  $0 \rightarrow 2$, $1 \rightarrow 0$, and $2 \rightarrow 1$, and lastly define $0$ as the identity map. The compositions of the maps will act as the group composition of the elements. Anyways, the construction of these maps are easy since $Z_{3}$ is cyclic. But how would I use the same concept for example on the group of reals under addition? Am I being to technical? Am I taking the definition of the morphism to literal? 
 A: 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.
A: A morphism is not necessarily a function from one object to another, this is merely the generic situation. All that is really required is that they can be composed. 
In any case, the maps that you're writing down are not weirdly technical, they are closely related to the notion of a group action. Given whatever group $G$, the category that encodes that group has a single group $G$ and its morphisms consist of all the functions $y \mapsto x \cdot y$ where $\cdot$ is the group operation and $x$ is an arbitrary group element. The composition is then just function composition.
