Straightforward translation of the modal formula using the definition of Kripke semantics leads to a monadic second-order $\Pi^1_1$ sentence. That’s about all that is possible, in general. There is no algorithm to find out whether the axiom is Kripke complete in the first place, whether it corresponds to a first-order condition, or if so, to compute the first-order condition. Even in simple concrete cases, these question often lead to very difficult problems.

Nevertheless, there are some classes of axioms that are better behaved. In particular, if the axiom happens to be equivalent to a Sahlqvist formula, it has a first-order equivalent, which can be effectively constructed.