This proof sketch of the second isomorphism theorem uses the symbols in my writeup there without repetition of the formulation, so you may want to read that first.
Define a homomorphism **f**: **G** -> **G**/**H** by **f**(*g*) = *g***H**. Then **f** is clearly an epimorphism. Since **N** is a subgroup of **H**, it's easy to see that the induced map **f**: **G**/**N** -> **G**/**H** defined by **f**(*g***N**) = *g***H** is a homomorphism. Now note that Ker **f** is precisely **H**/**N**, and that **f** is an epimorphism. The theorem follows from applying the first isomorphism theorem to **f**.