In this thesis, we analyse radiative transitions in charmonium using Lattice QCD, to provide insight into the internal structure and photocoupling of charmonium states. Distillation is used to allow the efficient computation of a high number of correlation functions, leading to the construction of highly optimized operators ideal as an input for the calculation of matrix elements. We explore techniques which leverage data over many time slices to increase statistics as well as a robust method of determining multiple form factors with generic kinematics. From these matrix elements, form factors are determined over a wide range of Q² and extrapolations are performed to the zero virtuality point. Quantities computed are in reasonable agreement with other lattice studies and experimental values, where available. First determinations of radiative partial widths of exotic charmonia using dynamical quarks are calculated.