We present new calculations of the photoionization of interstellar matter within ~5 pc of the Sun (which we refer to as the Complex of Local Interstellar Clouds or CLIC) by directly observed radiation sources including nearby hot stars and the diffuse emission of the Soft X-ray Background (SXRB). In addition, we model the important, unobserved EUV emission both from the hot gas responsible for the SXRB and from a possible evaporative boundary between the CLIC and the hot gas. We carry out radiative transfer calculations and show that these radiation sources can provide the ionization and heating of the cloud required to match a variety of observations. The ionization predicted in our models shows good agreement with pickup ion results, interstellar absorption line data towards epsilon CMa, and EUV opacity measurements of nearby white dwarf stars. Including the radiation from the conductive boundary improves agreement with data on the temperature and electron density in the cloud. The presence of dust in the cloud, or at least depleted abundances, is necessary to maintain the heating/cooling balance and reach the observed temperature. Using the column density observations as inputs, we derive the gas phase abundances of C, N, O, Mg, Si, S and Fe. Based on these inferred depletions, it appears that silicate and iron dust exists in the CLIC, while carbonaceous dust has been destroyed. ; Comment: 39 pages (incl. figures), 5 figures, AASTeX, accepted for publication in ApJ, some additional discussion and minor corrections added at request of the referee