A general approach is presented for spatially coarse-graining lattice kinetic Monte Carlo (LKMC) simulations of systems containing strongly interacting particles. While previous work has relied on approximations that are valid in the limit of weak interactions, here we show that it is possible to compute coarse-grained transition rates for strongly interacting systems without a large computational burden. A two-dimensional square lattice is employed on which a collection of (supersaturated) strongly interacting particles is allowed to reversibly evolve into clusters. A detailed analysis is presented of the various approximations applied in LKMC coarse graining, and a number of numerical closure rules are contrasted and compared. In each case, the overall cluster size distribution and individual cluster structures are used to assess the accuracy of the coarse-graining approach. The resulting closure approach is shown to provide an excellent coarse-grained representation of the systems considered in this study.