The book categories for the working mathematician, by Saunder's MacLane, comes well-recommended. Although I have never quite got around to reading it, it has such a promising title! Note that Saunder's MacLane was one of the original category theorists.
For an explicit example of graph theory and category theory working together to prove results on groups, there is a well-known link between the category of graphs and subgroups of free groups. The classical reference is Topology of Finite Graphs by Stallings. For example, if the non-diagonal component of the fibre product of a graph with itself is simply connected then the corresponding subgroup is malnormal. Dani Wise lifted this idea to the more general category of "cubical complexes", which allowed him to prove some famous open problems in group theory and G&T (for example, he proved the virtually Haken conjecture, and that every one-relator group with torsion is residually finite). I found Wise's paper The residual finiteness of positive one-relator groups to be especially helpful.
You say you want to use category theory and graphs to look at endomorphisms of groups. Well, one of the questions I was attacking in my PhD thesis was the following.
Fix a class of groups $\mathcal{C}$. Does every (countable) group occur as the outer automorphism group of a group from the class $\mathcal{C}$?
I fixed a class $\mathcal{C}$, and I managed to prove that the above result held for this class so long as I could prove that a certain group (in reality, a class of very similar groups) had a malnormal subgroup (with certain additional properties). So, I then took the fibre product of a "subgroup" in the ambient free group and (with a bit of effort) proved that its malnormality fell down to my group.
This doesn't quite answer your question, but I thought it relevant enough to mention. If you want, I can send you a copy of my thesis.