1)The weak topology $\sigma(E,E^*)$ on $E$ is defined as the weakest topology on $E$ which makes continuous the map $f:E\to\mathbb{R},$ for any $f\in E^*.$
2)The weak-* topology $\sigma(E^{**},E^*)$ on $E^{**}$ is by definition the weakest topology on $E^{**}$ which make continuous the map $J^*(f):\xi\in E^{**}\to\langle\xi,f\rangle\in\mathbb{R}$ for any $f\in E^*.$
Here I am considering the canonical map $J^*:E^*\to E^{***}$, starting on $E^*.$
3)A characteristic property of weak-* topology is that, for any topological space $(X,\tau),$ we have:
$F:(X,\tau)\to(E^{**},\sigma(E^{**},E^*))$ is continuous iff $J^*(f)\circ F:(X,\tau)\to\mathbb{R}$ is continuous, $\forall f\in E^*.$
Here and below I consider $\mathbb{R}$ with its usual topology.
4)By point 3) the canonical map $J:(E,\sigma(E,E^*))\to(E^{**},\sigma(E^{**},E^*))$ is continuous iff $J^*(f)\circ J:(E,\sigma(E,E^*))\to\mathbb{R}$ is continuous, $\forall f\in E^*.$
5)As you have noted, for any $f\in E^*,$ the map $J^*(f)\circ J:x\in E\to(J^*(f)\circ J)(x)\in\mathbb{R}$ coincides with $f:E\to\mathbb{R}$.
Finally by points 1), 4) and 5) you get the continuity of $J:(E,\sigma(E,E^*))\to(E^{**},\sigma(E^{**},E^*)).$
P.S. Certainly I could be more succint but I would highlight the single point or the argument. I hope it helps.