Hyper-Kamiokande will be a next-generation underground water Cherenkov detector with a total (fiducial) mass of 0.99 (0.56) million metric tons, approximately 20 (25) times larger than that of Super-Kamiokande. One of the main goals of Hyper-Kamiokande is the study of $CP$ asymmetry in the lepton sector using accelerator neutrino and anti-neutrino beams. In this paper, the physics potential of a long-baseline neutrino experiment using the Hyper-Kamiokande detector and a neutrino beam from the J-PARC proton synchrotron is presented. The analysis uses the framework and systematic uncertainties derived from the ongoing T2K experiment. With a total exposure of 7.5 MW $\times 10^7$ s integrated proton beam power (corresponding to $1.56 \times 10^{22}$ protons on target with a 30 GeV proton beam) to a $2.5^\circ$ off-axis neutrino beam, it is expected that the leptonic $CP$ phase $\delta _{CP}$ can be determined to better than 19 degrees for all possible values of $\delta _{CP}$ , and $CP$ violation can be established with a statistical significance of more than $3\,\sigma$ ( $5\,\sigma$ ) for $76{\%}$ ( $58{\%}$ ) of the ${\delta _{CP}}$ parameter space. Using both $\nu _e$ appearance and $\nu _\mu$ disappearance data, the expected 1 $\sigma$ uncertainty of $\sin ^2\theta _{23}$ is 0.015(0.006) for $\sin ^2\theta _{23}=0.5(0.45)$ .