We'll prove it in just one direction, since the other one is obvious. So, assume $\psi$ is a theorem of classical propositional logic. Prove that $\lnot \lnot \psi$ is a theorem of intuitionistic propositional logic.
My proof sketch is as follows. Assume $\lnot \lnot \psi$ is not a theorem of IPL. Then there exists a Kripke model (with a finite worlds count) such that within that model $$ \exists w : w \not \Vdash \lnot \lnot \psi, $$ implying $$ \exists u \geq w : u \Vdash \lnot \psi, $$ implying again $$ \exists u \geq w : \forall v \geq u : v \not \Vdash \psi. $$
Since the model we're in is finite, there exists some maximal world $\mu \geq u$, in which, by assumption, $\mu \not \Vdash \psi$. Since this world is maximal, the definitions of truthfulness for $\rightarrow, \lnot$ decay to the classical ones, and the values of the propositional variables in this world can naturally be viewed as a counterexample to $\psi$, which is a tautology and hence has no counterexamples! So, contradiction, and our original assumption is invalid.
Does this sound like a reasonable approach?
I can already prove the theorem by using (1) that $\lnot \lnot (\psi \lor \lnot \psi)$ is a theorem if IPL and (2) that if $\lnot \lnot \phi$ and $\lnot \lnot (\phi \rightarrow \psi)$ are theorems of IPL, then $\lnot \lnot \psi$ is too, so using these two facts I can rewrite any derivation of $\psi$ in CPL to a derivation of $\lnot \lnot \psi$ in IPL, but personally I find Kripke models more straightforward, hence the question.