Hybrid (organic shell-inorganic core) nanoparticles have important applications in nanomedicine. Although the inorganic components of hybrid nanoparticles can be characterized readily using conventional transmission electron microscopy (TEM) techniques, the structural and chemical arrangement of the organic molecular components remains largely unknown. Here, we apply TEM to the physico-chemical characterization of Au nanoparticles that are coated with plasma-polymerized-allylamine, an organic compound with the formula C3H5NH2. We discuss the use of energy-filtered TEM in the low-energy-loss range as a contrast enhancement mechanism for imaging the organic shells of such particles. We also study electron-beam-induced crystallization and amorphization of the shells and the formation of graphitic-like layers that contain both C and N. The resistance of the samples to irradiation by high-energy electrons, which is relevant for optical tuning and for understanding the degree to which such hybrid nanostructures are stable in the presence of biomedical radiation, is also discussed.