The use of short, synthetic DNA strands to mediate self-assembly of a collection of colloidal particles into ordered structures is now quite well established experimentally. However, it is increasingly apparent that DNA-linked colloidal assemblies (DLCA) are subject to many of the processing challenges relevant to atomic materials, including kinetic barriers related to nucleation and growth, defect formation, and even diffusionless transformations between different crystal symmetries. Understanding, and ultimately controlling, these phenomena will be required to truly utilize this technology to make new materials. Here, I describe a series of computational studies—based on a complementary suite of tools that includes Brownian dynamics, free energy calculations, vibrational mode theory, and hydrodynamic drag analysis—that address several issues related to the nucleation, growth, and stability of DNA-linked colloidal assemblies. The primary focus is on understanding the nature of the apparently enormous number of diffusionless solid-solid phase transformations that occur in crystallites assembled from DNA-functionalized colloidal particles. We find that the ubiquitous nature of these transformations is largely due to the short-ranged nature of DNA mediated interactions, which produces a panoply of zero-energy barrier pathways (or zero frequency vibrational modes) in a number of crystalline configurations. Furthermore, it is shown that hydrodynamic drag forces play a key role in biasing the transformations towards specific pathways, leading to unexpected order in the final arrangements. Additional studies also highlight how heterogeneity in the surface density of DNA strands grafted onto the particles may be used to improve nucleation and growth behavior, which is generally difficult in systems near the ‘sticky-sphere’ limit in which the interaction range is short relative to the particle size. In the final chapter of the thesis, a general and powerful technique is presented for extracting particle-particle interactions directly from particle trajectory data.