Let $C$ be a finitely axiomitisable class of $\Sigma$-structures for some given first-order signature sigma, ie there is a finite set $T$ of sentences whose models are precisely the members of $C$.
I want to show that the class of $\Sigma$-structures that are not in $C$ is also finitely axiomitisable.
I'm not sure if the following idea works. Given $T=\{\phi_1, \ldots, \phi_n\}$ let $\phi = \neg\phi_1 \lor\ldots\lor \neg\phi_n$, then the models of $\{\phi\}$ are precisely the members of $\overline C$ so $\overline C$ is finitely axiomitisable. Is this correct?
Yes, your reasoning is correct. Following is a motivation why it is correct.
If $M\not\models T$ then $M\not\models \phi_i$ for some $i$, thus $M\models \neg \phi_i$ and hence $M\models \neg \phi_i\vee ...\vee \neg \phi_n$.