The IceCube Neutrino Observatory at the South Pole is the world's largest neutrino detector with over 1km^3 of instrumented Antarctic ice. While it has been primarily designed to observe astrophysical neutrinos, this size also allows it to collect vast quantities of atmospheric neutrinos. These high-statistics datasets allow for measurements of the properties of neutrinos, in particular the phenomena of neutrino oscillation. One of the outstanding questions in this field is that of the neutrino mass ordering (NMO). The Precision IceCube Next Generation Upgrade (PINGU) is a proposed low-energy extension to IceCube for which a determination of the NMO is a priority science goal. The current low-energy atmospheric neutrino experiment at the South Pole, DeepCore, has been successfully collecting data since 2011. In this thesis the potential of this existing data to determine the NMO has been explored. While it was not expected to have a large sensitivity, this work has explored a Feldman-Cousins treatment for converting the delta-chi^2 between the two discrete mass ordering hypotheses into the standard Gaussian significance metric. Using 2.7 years of data from the DeepCore detector, the inverted mass ordering was preferred at the level of 0.05sigma. The second aspect of this thesis was to study the impact of the systematic uncertainties on the NMO determination. This particular analysis was actually statistics-limited and so the only impactful systematic uncertainties were the parameters that govern atmospheric neutrino oscillations, theta_23 and Deltam^2_31. Therefore, to improve the NMO results, these parameters were constrained by including the global information on them in the fits, yielding a new NMO sensitivity of 0.29sigma. This new global fit also yields measurements of the oscillation parameters of Deltam^2_32,NO=(2.443+/-0.037)e-3eV^2 and sin^2theta_23,NO=0.442+0.026-0.018 for the hypothesis of the normal mass ordering and Deltam^2_32,IO=(-2.510+/-0.036)e-3eV^2 and sin^2theta_23,IO=0.579+0.019-0.021 for the hypothesis of the inverted mass ordering. In addition to the work on the neutrino mass ordering, this thesis also investigated two issues related to predictions of the flux of atmospheric particles. The first related to the treatment of the predictions of the atmospheric neutrino flux, provided in binned tables. Crucially, these contain values representative of the integral of the flux across that bin and so an integral-preserving interpolation must be used. One such method will be presented along with a discussion of how it performs in the two-dimensional case of the atmospheric neutrino flux. The second issue related to quantifying uncertainties on the background muon distributions observed with the IceCube detector coming from the uncertainties on the initial cosmic ray flux. This involved performing a global fit on the available cosmic ray flux measurements and then propagating these uncertainties in to the muon distributions. To finalise this section, the exact manner in which these uncertainties can be included in the physics analyses of IceCube will be discussed.