The electrostatics of a simple membrane model picturing a lipid bilayer as a low dielectric constant slab immersed in a homogeneous medium of high dielectric constant (water) can be accurately computed using the exact Green's functions obtainable for this geometry. We present an extensive discussion of the analysis and numerical aspects of the problem and apply the formalism and algorithms developed to the computation of the energy profiles of a test charge (e.g., ion) across the bilayer and a molecular model of the acetylcholine receptor channel embedded in it. The Green's function approach is a very convenient tool for the computer simulation of ionic transport across membrane channels and other membrane problems where a good and computationally efficient first-order treatment of dielectric polarization effects is crucial.