Cell migration is one of the essential mechanisms responsible for complex biological processes. Intensive researches have begun to elucidate the mechanisms and search intriguing conditions for efficient control of cell migration. One general mechanism that is widely applicable for cells including Escherichia coli, amoebae and endothelial cell is chemotaxis. The single cell study for bacterial chemotaxis has an advantage over studies with the population of cells in providing a clearer observation of cell migration, which leads to more accurate assessments of chemotaxis. In this paper, we propose a three-dimensional model considering a single bacterium to study its chemotaxis. The semi-implicit Fourier spectral method is applied for high efficiency and numerical stability. The simulation results reveal rich dynamics of cell migration and provide quantitative assessments of bacterial chemotaxis with various chemoattractant gradient fields.