The aggregation-volume-bias Monte Carlo based simulation technique, which has led to our recent success in vapor-liquid nucleation research, was extended to the study of crystal nucleation processes. In contrast to conventional bulk-phase techniques, this method deals with crystal nucleation events in cluster systems. This approach was applied to the crystal nucleation of Lennard-Jonesium under a wide range of undercooling conditions from 35% to 13% below the triple. point. It was found that crystal nucleation in these model clusters proceeds initially via a vapor-liquid like aggregation followed by the formation of crystals inside the aggregates. The separation of these two stages of nucleation is distinct except at deeper undercooling conditions where the crystal nucleation barrier was found to diminish. The simulation results obtained for these two nucleation steps are separately compared to the classical nucleation theory (CNT). For the vapor-liquid nucleation step, the CNT was shown to provide a reasonable description of the critical cluster size but overestimate the barrier heights, consistent with previous simulation studies. On the contrary, for the crystal nucleation step, nearly perfect agreement with the barrier heights was found between the simulations and the CNT. For the critical cluster size, the comparison is more difficult as the simulation data were found to be sensitive to the definition of the solid cluster, but a stringent criterion and lower undercooling conditions generally lead to results closer with the CNT. Additional simulations at undercooling conditions of 40% or above indicate a nearly barrierless transition from the liquid to crystalline-like structure for sufficiently large clusters, which leads to further departure of the barrier height predicted by the CNT from the simulation data for the aggregation step. This is consistent with the latest experimental results on argon that show an unusually large underestimation of the nucleation rate by the CNT toward deep undercooling conditions.