Let's assume the characteristic is $0$.
Let $R$ be a normal $N$-graded ring of dimension 2 with homogenous maximal ideal $m$. Then $R$ has rational singularity if and only if the non-negative degrees part of the graded local cohomology module $H_m(R)$ vanish. That is $H_m(R)_{(i)}=0$ for $i\geq 0$. This is due to Watanabe, see Theorem 2.2 in: http://www.ams.org/tran/2003-355-03/S0002-9947-02-03186-0/home.html
It is safe to work in affine situation, so let $S=k[x,y]$ and $R=S^G$. Then $R$ is normal and generated by forms of positive degree. Since $H_{(x,y)}(S)_{(i)}=0$ for $i\geq 0$, it follows that $H_m(R)_{(i)}=0$ for $i\geq 0$ (one can compute local cohomology in $S$ by using a system of parameters which are elements in $R$).
In characteristic $p$ one probably has to use Frobenius. Note that Boutot's theorem fails in this case (I think it is still true for finite group though).
A truly easy proof is probably not easy to find unless one has a truly elementary definition of rationality.