We present a novel, biomimetic model of the eye for realistic virtual human animation. We also introduce a deep learning approach to oculomotor control that is compatible with our biomechanical eye model. Our eye model consists of the following functional components: (i) submodels of the 6 extraocular muscles that actuate realistic eye movements, (ii) an iris submodel, actuated by pupillary muscles, that accommodates to incoming light intensity, (iii) a corneal submodel and a deformable, ciliary-muscle-actuated lens submodel, which refract incoming light rays for focal accommodation, and (iv) a retina with a multitude of photoreceptors arranged in a biomimetic, foveated distribution. The light intensity captured by the photoreceptors is computed using ray tracing from the photoreceptor positions through the finite aperture pupil into the 3D virtual environment, and the visual information from the retina is output via an optic nerve vector. Our oculomotor control system includes a foveation controller implemented as a locally-connected, irregular Deep Neural Network (DNN), or "LiNet", that conforms to the nonuniform retinal photoreceptor distribution, and a neuromuscular motor controller implemented as a fully-connected DNN, plus auxiliary Shallow Neural Networks (SNNs) that control the accommodation of the pupil and lens. The DNNs are trained offline through deep learning from data synthesized by the eye model itself. Once trained, the oculomotor control system operates robustly and efficiently online. It innervates the intraocular muscles to perform illumination and focal accommodation and the extraocular muscles to produce natural eye movements in order to foveate and pursue moving visual targets. We additionally demonstrate the operation of our eye model (binocularly) within our recently introduced sensorimotor control framework involving an anatomically-accurate biomechanical human musculoskeletal model.