The McKay correspondence states that there is a bijection between the McKay graphs of finite subgroups of $SU_2$ and Dynkin diagrams in the $ADE$ classification system of simply laced Lie algebras. We investigate this correspondence by finding the finite subgroups of $SU_2$ and explicitly constructing the McKay graph corresponding to each group. We then view these groups from the lens of algebraic geometry and go through the blow-up procedure for the corresponding Kleinian singularities to demonstrate that the blow-up graph, quite remarkably, yields another way to obtain the Dynkin diagrams.