The title should explain my question thoroughly enough. I can't seem to get started anywhere. Intuitively it seems like some kind of brute-forcing would work i.e if the DFA has the symbols $\Sigma$ you would simply be passing strings from $\Sigma^*$ until the DFA rejects. What would even be the time complexity of this though? Or am I completely out and about?
The problem with your approach is that there are infinitely many strings in $\Sigma^*$. If the automaton keeps accepting and accepting, how would you know hen to stop?
Instead, consider how an automaton can reject: it rejects by getting to a rejecting state.