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We develop several new algorithms using molecular simulation to investigate the nucleation barrier of a single, freely-jointed polymer chain. In the first part of the thesis, we use a free particle model to develop a new biasing technique, which uses an automated feedback mechanism to overcome the poor sampling of crystal states in a thermodynamic system. Our feedback technique does not require any prior knowledge of the nucleation barrier and enables good representative sampling of all available states of interest. In the second part of the thesis, we simulate the nucleation barrier of the single, freely-jointed, square-well chain. We use our feedback technique and parallel tempering with a nonstandard temperature distribution to overcome poor sampling of crystal states and configuration mixing issues respectively. We also provide some comparative analysis of different choices of configurational order parameters for the single chain. Finally, we apply stretching to the chain to approximate flow-induced crystallisation and investigate the effect of different degrees of stretch on the nucleation barrier. We verify the quality of our simulation with careful monitoring of several criteria, including the acceptance ratios of configuration swaps between simulations with adjacent temperatures, evolution of the energy traces as a result of configuration swaps between tempering levels, and ensuring effective de-correlation of configurations through reptation moves. Our simulations provide strong reproducible results for the base, the peak and beyond the peak of the barrier for the quiescent and stretched single chain. We observe a remarkably strong effect of modest stretching on the nucleation barrier for a single chain, which can potentially lead to dramatic effects on the nucleation rate. Our simulation code has been made publicly available, with details provided in an appendix.