We investigate the interfacial structure of ionic solutions consisting of alkali halide ions in water at concentrations in the range 0.2-1.0 molal and at 300 K. Combining molecular dynamics simulations of point charge ion models and a recently introduced computational approach that removes the averaging effect of interfacial capillary waves, we compute the intrinsic structure of the aqueous interface. The interfacial structure is more complex than previously inferred from the analysis of mean profiles. We find a strong alternating double layer structure near the interface, which depends on the cation and anion size. Relatively small changes in the ion diameter disrupt the double layer structure, promoting the adsorption of anions or inducing the density enhancement of small cations with diameters used in simulation studies of lithium solutions. The density enhancement of the small cations is mediated by their strong water solvation shell, with one or more water molecules "anchoring" the ion to the outermost water layer. We find that the intrinsic interfacial electrostatic potential features very strong oscillations with a minimum at the liquid surface that is ∼4 times stronger than the electrostatic potential in the bulk. For the water model employed in this work, SPC/E, the electrostatic potential at the water surface is ∼-2 V, equivalent to ∼80 k(B)T (for T = 300 K), much stronger than previously considered. Furthermore, we show that the utilization of the intrinsic surface technique provides a route to extract ionic potentials of mean force that are not affected by the thermal fluctuations, which limits the accuracy of most past approaches including the popular umbrella sampling technique.