An interfacial instability driven by concentration diffusion in a solid system is studied using simulations based on Ising lattice-gas energetics with conserved dynamics. Interfacial thermodynamic properties including the effect of surface tension are included at the microscopic level. No additional modeling of interfacial boundary conditions is required. Finger structures evolve from a flat initial interface and are studied on moderate time scales, well into the nonlinear regime. The dynamics and universality of the growth process are studied. Evidence is presented for power-law growth and for a scaling (self-similar) form in the evolution of the Fourier transform of the interface shape. A scaling law relating two exponents appearing in the scaling form is presented; it is based on the "inverse cascade" of energy introduced on the shortest length scales and evolving to the long scales of the interfacial structure. In some ways the simulations and scaling presented are the analog, for the case of interfacial instabilities, of the scaling laws observed in simulations and experiments on spinodal decomposition. Most of the results presented will apply to the symmetric two-sided system; preliminary results for the "one-sided" asymmetric case, which appears to lie in a different universality class, will also be discussed.