Quotient isomorphism question. 
I'm confused about where the equivalence relations come in and how they are related to the maps. 
 A: Presumably $\sim_{\varphi}$ is the equivalence relation $x\sim_{\varphi} y$ iff $\varphi(x)=\varphi(y)$, and similarly for $\sim_\pi$. $\sim_N$ should be the equivalence relation $x\sim_N y$ iff $x^{-1}y \in N$. You can then proceed as the question indicates.
A: The equivalence relation comes into play when proving that the map $\tilde \varphi(aN) = \varphi(a)$ is well-defined – independent of the choice of the element $a$ from its equivalence class. That is, if $a \sim_N b$ then $\varphi(a) = \varphi(b)$, so you can define a function $\tilde \varphi$ on $G/N$ whose value at any equivalence class $[a]_N$ is the common value of $\varphi$ on $[a]_N$.
The equivalence relation also comes into play when showing that $\tilde \varphi$ is injective: by definition, elements in different equivalence classes have different values under $\varphi$, so those equivalence classes have different values under $\tilde \varphi$.
Clearly $\tilde \varphi$ is onto $\overline G$, as $\varphi$ is. 
Actually, there's only one equivalence relation involved: $\sim_{\varphi}, \sim_{\pi}, \sim_N$ are identical.
