Take ring $R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} | \, a \equiv b\, \mod 5 \}$. Find all ring homomorphism $R \rightarrow \mathbb{C}$ I take generators of $R$ to be $(1,1)$ and $(0,5)$. I have following:
$\phi((0,0))=0$ and 
$\phi((1,1))=1$. 
If $\phi((0,5))=a$ then $a-5=\phi((0,5)-5(1,1))=\phi((-5,0)$, and so:
$a(a-5)= \phi((0,5))\phi((-5,0))= \phi((0,5)\cdot(-5,0))=\phi((0,0))=0$
Hence $\phi((0,5))=0$ or $5$. 
Are those all homomorphisms?
 A: Ring homomorphism is not a term that all mathematicians agree about. Some mathematicians define a ring homomorphism to be a function $\phi:R_1\to R_2$ such that $\phi(x+y)=\phi(x)+\phi(y)$ and $\phi(x\cdot y)=\phi(x)\cdot\phi(y)$ for all $x,y\in R_1$, and some mathematicians insist that a ring homomorphism $\phi:R_1\to R_2$ must also satisfy $\phi(1_{R_1})=1_{R_2}$. Mathematicians typically decide whether to include this extra condition based on what is most convenient for their research.
If you feel that a ring homomorphism $\phi:R_1\to R_2$ must also satisfy $\phi(1_{R_1})=1_{R_2}$, then you got them all.
If you do not require that ring homomorphisms $\phi:R_1\to R_2$ satisfy $\phi(1_{R_1})=1_{R_2}$, then there is one additonal ring homorphism: the map $R\to\mathbb{C}$ which sends $x\mapsto0$ for all $x\in R$.
Everything you said in your post was correct. If you're not sure you should believe what you've done, then here are some exercises you could do to help you make sure you understand everything. You'll probably be able to do all of these on your own with no problem, but if you'd like solutions to these exercises, then let me know. I'll be happy to help.
Exercise 1: Let $\phi:R_1\to R_2$ be a ring homomorphism, and let $y=\phi(1)$. Show that $y^2=y$. If $R_2$ is an integral domain, then show that $y=0$ or $y=1$. If $y=0$, show that $\phi(x)=0$ for all $x\in R_2$.
Exercise 2: Let $R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} | \, a \equiv b\text{ mod }5 \}$. Show that for every $(a,b)\in R$, there exist unique $c,d\in\mathbb{Z}$, such that $(a,b)=c(1,1)+d(0,5)$.
You mentioned that $(1,1)$, $(0,5)$ are generators. The point of the previous exercise is to note that every element of $R$ can be expressed uniquely in terms of these generators.
Exercise 3: Let $R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} | \, a \equiv b\text{ mod }5 \}$, and define $\phi_1,\phi_2:R\to\mathbb{C}$ by letting
$$\phi_1\left(a(1,1)+k(0,5)\right)=a$$
and
$$\phi_2\left(a(1,1)+k(0,5)\right)=a+5k$$
for all $a,k\in\mathbb{Z}$. Show that $\phi_1,\phi_2$ are ring homomorphisms.
With the above exercises, and the work you've done in the original post, you can conclude that the ring homomorphisms $R\to\mathbb{C}$ are $\phi_1$ and $\phi_2$, (and maybe the map sending each $x\mapsto0$, as well).
