Thesis (Ph.D.) - Indiana University, Department of Physics, 2018The search for neutrinoless double beta decay $(0v\beta\beta)$, a rare process which may reveal the Dirac/Majorana nature of the neutrino, requires a careful understanding of all backgrounds. A precise study of neutron interactions on $^{136}$Xe, a common isotope in double beta decay experiments, is necessary for the background models in experiments such as EXO-200 and, in the future, nEXO. Neutron capture and neutron inelastic scattering on $^{136}$Xe have been studied at the Detector for Advanced Capture Experiments (DANCE) and the GErmanium Array for Neutron Induced Excitations (GEANIE) at the Los Alamos Neutron Science Center. The relative neutron cross sections for neutron capture at thermal and first resonance energies have been measured. The gamma cascades for these captures have also been measured, and cascade models have been developed. Partial gamma ray cross sections of (n,xn$\gamma$) are being measured with data from GEANIE. These data will also be used to set limits on the number of events in the $0v\beta\beta$ ROI for $^{136}$Xe. These studies will lead to improved sensitivity to neutrinoless double beta decay in the future.