The relativistic many-body perturbation theory with the optimized Dirac-Kohn-Sham zeroth approximation is applied to calculation of the hyperfine structure parameters for some Li-like multicharged ions. The relativistic, exchange-correlation and other corrections are accurately taken into account. The optimized relativistic orbital basis set is generated in the optimal many-body perturbation theory approximation with fulfilment of the gauge invariance principle. The obtained data on the hyperfine structure parameters of the Li-like multicharged ions are analyzed and compared with alternative theoretical and experimental results