$X$ is a compact complex manifold of dimension $n$ (Not necessarily Kahler). How would I go about proving the statement above? I know that $H^n(X,K_X)$ is isomorphic to $H^{n,n}(X)$ by Dolbeault's theorem.
Integration gives a well defined map onto $\mathbb{C}$ but I'm not sure how to prove that it's one-one.
The isomorphism $H^n(X,K_X)\simeq\mathbb{C}$ follows from Serre duality, which is valid for every compact complex manifold, Kähler or not.
One version of Serre's fundamental result is that given a connected complex compact manifold $X$ of dimension $n$, a vector bundle $V$ on $X$ and an integer $p$, the finite-dimensional vector spaces $$H^p(X,V) \operatorname {and} H^{n-p}(X,K_X\otimes V^*)$$ ( which are finite-dimensional by Cartan-Serre!) are canonically dual for any integer $p$.
Your isomorphism is obtained by taking $p=n$ and $V=K_X.$
Bibliography
Serre, Un théorème de dualité