This work develops a stochastic model to explore the dynamical behaviour of hepatitis B virus (HBV) infection by incorporating virus-to-cell transmission, cell-to-cell transmission, and the cytotoxic T lymphocyte (CTL) immune response under Lévy noise, which represents sudden environmental perturbations. The model is formulated, and its feasibility is investigated. The study is further enriched by proving the uniqueness, existence, and positivity of the global solution. A sufficient extinction condition is derived by using a combined infected-virus Lyapunov functional. In addition, a conditional sufficient criterion for long-term average persistence is established under bounded-state dynamics. The extinction result shows that infected hepatocytes and free virus particles vanish when the corresponding sufficient extinction condition is satisfied, whereas persistence is obtained under the boundedness assumption on the normalization factor and the condition ${\mathrm{\Delta }}_P>0$. At the end, numerical simulations are performed to validate and illustrate the analytical findings. Additional numerical reliability tests, including step-size sensitivity, relative error estimation, positivity verification, and comparison with a positivity-preserving truncated Euler-Maruyama scheme, are also provided. The results demonstrate that stochastic environmental perturbations can significantly affect the transmission of infectious diseases, including HBV. Significantly, excessive noise levels can greatly restrict a population's ability to spread the disease.