All-inorganic halide double perovskites have emerged as a promising class of materials that are potentially more stable and less toxic than lead-containing hybrid organic-inorganic perovskite optoelectronic materials. In this work, 311 cesium chloride double perovskites (Cs2BB'Cl-6) were selected from a set of 903 compounds as likely being stable on the basis of a statistically learned tolerance factor (t) for perovskite stability. First-principles calculations on these 311 double perovskites were then performed to assess their stability and identify candidates with band gaps appropriate for optoelectronic applications. We predict that 261 of the 311 Cs2BB'Cl-6 compounds are likely synthesizable on the basis of a thermodynamic analysis of their decomposition to competing compounds (decomposition enthalpy <0.05 eV/atom). Of these 261 likely synthesizable compounds, 47 contain no toxic elements and have direct or nearly direct (within 100 meV) band gaps between 1 and 3 eV, as computed with hybrid density functional theory (HSE06). Within this set, we identify the triple-alkali perovskites Cs-2[Alk](+)[TM]Cl-3+(6), where Alk is a group 1 alkali cation and TM is a transition-metal cation, as a class of Cs2BB'Cl-6 double perovskites with remarkable optical properties, including large and tunable exciton binding energies as computed by the GW-Bethe-Salpeter equation (GW-BSE) method. We attribute the unusual electronic structure of these compounds to the mixing of the Alk-Cl and TM-Cl sublattices, leading to materials with small band gaps, large exciton binding energies, and absorption spectra that are strongly influenced by the identity of the transition metal. The role of the double-perovskite structure in enabling these unique properties is probed through an analysis of the electronic structures and chemical bonding of these compounds in comparison with other transition-metal and alkali transition-metal halides.