We introduce a model for the explicit evolution of interstellar dust in a cosmological galaxy formation simulation. We post-process a simulation from the Cosmic Reionization on Computers project \citep[CROC,][]{Gnedin2014}, integrating an ordinary differential equation for the evolution of the dust-to-gas ratio along pathlines in the simulation sampled with a tracer particle technique. This model incorporates the effects of dust grain production in asymptotic giant branch star (AGB) winds and supernovae (SN), grain growth due to the accretion of heavy elements from the gas phase of the interstellar medium (ISM), and grain destruction due to thermal sputtering in the high temperature gas of supernova remnants (SNRs). A main conclusion of our analysis is the importance of a carefully chosen dust destruction model, for which different reasonable parameterizations can predict very different values at the $\sim 100$ pc resolution of our simulations. We first test this dust model on the most massive galaxy in a 10$h^{-1}$ co-moving megaparsec (Mpc) box, for which we find that the total predicted dust mass is somewhat sensitive to parameter choices for the dust model, especially the timescale for grain growth due to accretion in the ISM. To test whether dust-dependent observable quantities of galaxies at these epochs could be useful for placing constraints on dust physics, we then apply the model to a suite of 11 simulated galaxies with stellar masses from $\sim 10^5 - 10^9 M_{\odot}$ in the first $1.2$ billion years of the universe to make predictions for the dust content of high-redshift galaxies. We explore 9 different sets of dust model parameters, forward modelling observable properties of high-redshift galaxies to compare to data. We find that we are unable to simultaneously match existing observational constraints with any one set of model parameters. Specifically, the models which predict the largest dust masses $D/Z \gtrsim 0.1$ at $z = 5$ -- because of high assumed production yields and/or efficient growth via accretion in the ISM -- are preferred by constraints on total dust mass and IR luminosities, but these models produce far too much extinction in the UV, preventing them from matching observations of $\beta_{\rm UV}$. To investigate this discrepancy, we analyze the relative spatial distribution of stars and dust as probed by infrared (IR) and ultraviolet (UV) emission. We find that all models predict significant dust attenuation in the central region of the galaxy, resulting in a ring-like morphology for the UV emission. Since IR emission peaks in the center of the galaxy, there are $\sim$ kpc-scale offsets between the points of maximal UV and IR surface brightness when ``observed'' with infinite resolution, but degrading image resolution to be similar to existing observational capabilities results in no offset between peak brightness in UV and IR. While existing observations only probe galaxies brighter in the UV than the most massive in our sample, they do exhibit much larger offsets that are suggestive of more complicated morphologies. Our results therefore provide strong motivation for the development of a dust model such as the one presented in this dissertation in higher-resolution simulations of galaxy formation which more realistically reproduce the dynamics of the reionization-era ISM.