Deducing weak convergence of $f_n$ s.t for every $g \in L^q(\mu)$, $\lim_n \int f_n g \,d\mu$ converges

If $\phi$ is a bounded linear functional on $L^q(\mu)$ then there exists $f \in L^p(\mu)$ with $\phi(g) = \displaystyle \int fg \, d\mu$. You don't have to find it; Riesz did that for you.

Under your definition of $\phi$ this means $$\lim_n \int f_ng \, d\mu = \int fg \, d\mu$$ for all $g \in L^q(\mu)$, verifying weak convergence.

Clearly, $\phi(g)=\lim \int_X f_ng\,d\mu$ is well-defined and linear. Moreover, $$|\phi(g)|\leq M\|g\|_q$$ so $\|\phi\|\leq M$.

By Riesz Representation, there exists a unique $f\in L^P$ such that $\phi(g)=\int_X fg\,d\mu$. But this means $\lim \int_X (f_n-f)g \,d\mu=0.$