A free boundary problem for the chemotaxis model of parabolic-elliptic type is investigated in the present paper, which can be used to simulate the dynamics of cell density under the influence of the nonlinear diffusion and nonlocal attraction-repulsion forces. In particular, it is shown for supercritical case that if the initial total mass of cell density is small enough or the interaction between repulsion and attraction cancels almost each other, the strong solution for the cell density exists globally in time and converges to the self-similar Barenblatt solution at the algebraic time rate, and for subcritical case that if the initial data is a small perturbation of the steady-state solution and the attraction effect dominates the process, the strong solution for cell density exists globally in time and converges to the steady-state solution at the exponential time rate.