We investigate the nonlinear emission dynamics of quantum dots coupled to photonic crystal cavities in the Purcell regime using luminescence intensity autocorrelation. Two laser pulses with a controlled time delay sequentially excite the coupled system inducing emission that depends on the delay and laser power. We find distinct contrasts between exciton and biexciton emission as a function of time delay which originate from different nonlinearities. A quantum optical simulation is also performed that accounts for the interaction between the laser pulses, exciton, and cavity mode.