We study the dynamics of vortex rings in Bose-Einstein condensates (BECs), the coldest known matter in the universe. For appropriate conditions (dilute enough and low enough temperatures), a BEC behaves as a quantum superfluid modeled by the nonlinear Schrödinger equation. Using the Bio-Savart law for standard fluids, we show how the interactions between coaxial vortex rings in the 3D superfluid can be approximated by coupled particle equations on their positions and radii. In particular, we study the leapfrogging behavior of coaxial vortex rings with equal charge using the particle model and compare it favorably to the full 3D dynamics of the original BEC model. Conditions for the existence of leapfrogging behavior are studied, as well as its stability. An extension of a GPU-accelerated code is developed to study the dynamics of vortex rings confined in a parabolic magnetic parabolic trap. In particular, we focus on the evolution of unstable stationary states, vortex ring precession around stable stationary states, and leapfrogging behavior.