To study the precise mathematical formulation of path integrals, you actually need probabilistic tools. The path integral is a stochastic integral with suitable measures, such as the Wiener measure associated with brownian motion.

The ideas used by physicists are very useful, but not always mathematically accurate, and rely more or less on justification by approximation of these stochastic integrals.

So it really depends on the purpose for you to study them. If you plan to utilize them as a physicist do (or just to understand their physical meaning), not much mathematical background will be needed, apart from basic quantum mechanics knowledge (and Trotter formula) and the principle of least action of classical mechanics (a bit of calculus of variations, as already mentioned).

But if you are interested in a more rigorous study, perhaps oriented to mathematical physics, you really need to understand stochastic processes and probability.