Relativistic nonthermal plasmas are ubiquitous in high-energy astrophysical systems featuring turbulence such as pulsar wind nebulae and active galactic nuclei, as inferred from broadband nonthermal emission spectra. The underlying turbulent nonthermal particle acceleration (NTPA) processes have traditionally been modelled with a Fokker-Planck (FP) diffusion-advection equation for the particle energy distribution. In this dissertation, I test FP-type NTPA theories by analysing three-dimensional (3D) particle-in-cell (PIC) simulations of magnetised turbulence in collisionless relativistic pair plasma. By tracking the energy histories of large numbers of particles in several simulations with different initial magnetisation σ0 and system size, I first test the energy-diffusion assumption of the FP framework, finding simple diffusion throughout the parameter space. I then measure the FP energy diffusion and advection coefficients (D and A, respectively) as functions of particle energy γmc², and compare their dependence on initial and instantaneous system parameters to theoretical predictions. In the high-energy nonthermal tail, I find, robustly with respect to system size and σ₀, that D ~ γ², with a more complicated but generally shallower scaling at thermal and subthermal energies which varies qualitatively depending on σ₀. Hence, I fit D = D₀γ² in the nonthermal region and find that the scaling of D₀ with the instantaneous magnetisation σ(t) is consistent with D₀ ~ σ^3/2, although this flattens somewhat at higher σ ~ 1. I also measure the evolution of the power-law index α(t) of the particle energy distribution and find that it is well-described by an exponential convergence in time. I then build and test an analytic model connecting the FP coefficients and the observed power-law evolution, predicting that A(γ) ~ γ log(γ/γ*_A). This is consistent with my measurements of A(γ, t), and I furthermore find that the measured A(γ, t) can acceptably predict α(t) through the model relations. These results suggest that the basic 2nd-order Fermi acceleration model, which predicts D₀ ~ σ, may not be a complete description of NTPA in turbulent collisionless relativistic plasmas. My findings encourage further application of tracked particle methods and FP coefficient measurements as a diagnostic in kinetic simulations of various physical situations including collisionless shocks and magnetic reconnection, with relevance to astrophysical plasmas.