In this work, we study the onset of nonlinear rheological behavior of a colloidal dispersion of a synthetic hectorite clay, LAPONITE (R), at the critical gel state while undergoing the sol-gel transition. The critical gel state is characterized by a power-law dependence of stress relaxation modulus on time and structurally represents the weakest space spanning percolated fractal network. When subjected to step strain in the nonlinear regime, the relaxation modulus shifts vertically to the lower values such that the deviation from linearity can be accommodated using a strain dependent damping function. We also perform creep-recovery and start-up shear experiments on the studied colloidal dispersion at the critical gel state and monitor deviation in response as the flow becomes nonlinear. A quasilinear integral model is developed with the time-strain separable relaxation modulus to account for the effect of nonlinear deformation. Remarkably, the proposed model predicts the deviation from linearity in the creep-recovery and start-up shear experiments very well, leading to a simple formulation to analyze the onset of nonlinear rheological behavior in the critical gels. We also analyze the energy dissipation during the nonlinear deformation. We show that, for a critical gel, if the onset of nonlinearity happens to be the point of failure, the Boltzmann superposition principle very naturally leads to the Basquin law of failure.