The Dirac-Hartree-Fock approximation is adopted to calculate the mass shift and the field shift for the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition in Hg⁺. It is found that the field shift is much larger than the mass shift so that the latter can be neglected in the isotope shift. In addition, we estimate that the isotope shifts of the levels related to the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition of Hg⁺ is on the order of about 10⁴ GHz, while the hyperfine structure splitting is in a range of 1-10 GHz. However, the isotope shift of the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition is on the same order of magnitude as the hyperfine structure splitting. Therefore, the hyperfine structure splitting must be taken into account for predicting the frequency shifts of the clock transition between different isotopes. On the basis of these results, we perform a multi-configuration Dirac-Hartree-Fock calculation on the field shift of the 5d¹⁰6s ²S_1/2→5d⁹6s^{2 2}D_5/2 clock transition in Hg⁺ and the hyperfine interaction constants of the upper and the lower levels involved. In order to give accurate theoretical results of these physical quantities, we systematically consider the main electron correlations in the atomic system by using the active space method. The restricted single and double (SrD) excitation method is used to capture the correlation between the 5d and the 6s valence electrons, and the correlation between the 3s, 3p, 3d, 4s, 4p, 4d, 5s, 5p, and 5d core and the valence electrons. The isotope shifts and hyperfine structure splitting for this transition of several stable mercury isotopes are given. In particular, the uncertainty of the calculated isotope shift between ¹⁹⁹Hg⁺ and ¹⁹⁸Hg⁺ is about 2%, compared with the experimental measurement available. Using these results, we predict the absolute frequency values of this transition for seven mercury isotopes, which provides theoretical reference data for experiments. Moreover, the calculated isotope shifts and hyperfine structures are also useful for studying the structure, property and nucleon interaction of mercury nucleus.