I have a Hausdorff separated locally convex barreled space $(X,\tau)$ with topological dual $X^*$. My questions are:
$Q_1$ Is there a topology $\tau^*$ on $X^*$ that is finer than the weak-star topology $w^*$ such that $(X^*,\tau^*)$ is Frechet?
$Q_2$ If the answer for $Q_1$ is negative what extra conditions need be fulfilled so that $Q_1$ gets a positive answer?
The answer is quite often negative. For example, if $(X,\tau)$ is itself a Frechet space, then there is a Frechet topology on $X^*$ finer than the weak*-topology if and only if $X$ is Banach.
For the proof one considers the so-called inductive topology on $X^*$ (which is the inductive topology w.r.t. all embeddings of the Banach spaces whose unit balls are the closed absolutely convex equi-continuous sets) and applies the closed graph theorem to see that the Frechet topology is finer than the inductive topology. Then the open mapping theorem shows that those topologies are in fact equal. Finally, Baire's theorem implies that there is a single equi-continuous set which absorbs all others and this implies that $X$ is normed (hence Banach).
To have the dual of the barrelled space $X$ Frechet you need that $X$ has a fundamental sequence of bounded sets, that is $(B_n)_n$ is a sequence of bounded sets such that every bounded set is contained in some $B_n$.
This is part of Grothendieck's definition of (DF)-spaces (which need not be barrelled but only "countably quasi-barrelled").