Ultracold neutral plasmas are created by photoionizing laser-cooled atoms in a magneto-optical trap (MOT). Due to their large electrical potential energies and comparatively small kinetic energies, ultracold plasmas fall into a regime of plasma systems which are called strongly coupled. A priority in the field of ultracold plasmas is to generate plasmas with higher values of the strong coupling parameter Î“, which is given as the ratio of the nearest-neighbor Coulomb potential energy to the average kinetic energy. The equilibrium strong coupling in ultracold plasmas is limited by the ultrafast relaxation of the ions due to spatial disorder in the initial system. This heating mechanism is called â€œdisorder-induced heatingâ€ (DIH) and it limits the ion strong coupling in ultracold plasmas to order unity. This thesis describes experiments that explore ways to generate higher values of the strong coupling parameter in an ultracold neutral calcium plasma.One way to increase Î“ is to mitigate the effects of DIH using electron screening. This thesis describes an experiment in which the initial electron temperature was systematically changed to determine the effect that electron screening has on the ion thermalization. At lower initial electron temperatures, corresponding to a higher degree of electron shielding, it was found that the screening slows the ion thermalization and reduces the equilibrium ion temperature by as much as a factor of two. However, electron screening also reduces the ion interaction strength by the same amount, which has the net effect of leaving the effective Î“ unchanged.Another method for increasing the strong coupling of an ultracold plasma is to excite the plasma ions to a higher ionization state. Simulations predict that doubly ionizing the plasma ions can increase the strong coupling in an ultracold plasma by as much as a factor of 4, with the maximum value of Î“ depending on the timing of the second ionization relative to the DIH process. This thesis describes an experiment designed to test these predictions in a CaÂ²âº plasma. Measurements of the change in the Caâº ion temperature as a function of the timing of the second ionization pulses were made using laser-induced fluorescence. Results of these measurements show that the heating of the Caâº ions due to the second ionization depends on the timing of the second ionization pulses, as predicted by MD simulations.