Let $x_0$ be the point of $P^2 \vee P^2$ where we glued the two copies of $P^2$ together.
Recall that to apply Van Kampen theorem (and obtain an isomorphism), we need to write $P^2 \vee P^2$ as the union of path-connected open sets, each containing $x_0$, such that the intersection of any three is path-connected. We should aim to write $P^2 \vee P^2$ using only two path-connected open sets, as that will make our job much easier.
My hint would be to take advantage of the fact that you have two copies of $P^2$, which you know is path-connected (right?) and whose fundamental group you know (right?), after all you should want to write the fundamental group of $P^2 \vee P^2$ (what should your basepoint be?) in terms of some well-known group.