We study the approach to jamming in hard-sphere packings and, in particular, the pair correlation function g(2) (r) around contact, both theoretically and computationally. Our computational data unambiguously separate the narrowing delta -function contribution to g(2) due to emerging interparticle contacts from the background contribution due to near contacts. The data also show with unprecedented accuracy that disordered hard-sphere packings are strictly isostatic: i.e., the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom, once rattlers are removed. For such isostatic packings, we derive a theoretical connection between the probability distribution of interparticle forces P(f) (f) , which we measure computationally, and the contact contribution to g(2) . We verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state. We clearly observe a maximum in P(f) and a nonzero probability of zero force, shedding light on long-standing questions in the granular-media literature. We computationally observe an unusual power-law divergence in the near-contact contribution to g(2) , persistent even in the jamming limit, with exponent -0.4 clearly distinguishable from previously proposed inverse-square-root divergence. Additionally, we present high-quality numerical data on the two discontinuities in the split-second peak of g(2) and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features. Finally, we present the computational data on the contact contribution to g(2) for vacancy-diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings. We find that the contact network remains isostatic even when ordering is present. Unlike previous studies, we find that ordering has a significant impact on the shape of P(f) for small forces.