CALCLENS, written in C and employing widely available software libraries, efficiently computes weak gravitational lensing shear signals from large N-body light cone simulations over a curved sky. The algorithm properly accounts for the sky curvature and boundary conditions, is able to produce redshift-dependent shear signals including corrections to the Born approximation by using multiple-plane ray tracing, and properly computes the lensed images of source galaxies in the light cone. The key feature of this algorithm is a new, computationally efficient Poisson solver for the sphere that combines spherical harmonic transform and multgrid methods. As a result, large areas of sky (~10,000 square degrees) can be ray traced efficiently at high-resolution using only a few hundred cores on widely available machines. Coupled with realistic galaxy populations placed in large N-body light cone simulations, CALCLENS is ideally suited for the construction of synthetic weak lensing shear catalogs to be used to test for systematic effects in data analysis procedures for upcoming large-area sky surveys.