Of concern is the fundamental "minimal" chemotaxis model. It is known that when the chemotaxis coefficient is large enough, the system has a monotone, spiky steady state. We obtain the asymptotics of the steady state when the chemotaxis coefficient goes to infinity; we then use this fine information about the structure of the spike to prove its stabillity. Techniques involved to achieve these goals are singular perturbation method, asymptotic analysis of a nonlocal eigenvalue problem and a homotopy trick that links the nonlocal eigenvalue problem to the linearized eigenvalue problem of the chemotaxis system.