Proving Poisson's Equation

The ##r## on the lefthand side is the distance between two points whereas the ##r## in the argument of the delta function should be the vector ##\vec{r}##.

The potential V isn't a function of only radial distance. It's a function of all three coordinates. In my edition of Griffiths, he uses the notation V(P) to indicate it's the potential at point P. He doesn't write V(r), which implies something much different. The symbols ##p## and ##\rho## look similar, but they aren't the same symbol. Be consistent. You should also define exactly what you mean by ##r## inside the integral. How is it related to the variables of integration? Finally, ##dr'## should actually be the volume element ##d\tau##.

The lefthand side should not be the potential V.

Electromagnetism is a fairly math-intensive course, so you should really make an effort to be precise with your notation. The sloppiness that you could get away with in earlier courses will really start to hinder you now. Clean it up so you can avoid making dumb errors and confusing yourself.