The onset of nonlinear dynamics and chaos is evidenced in a mid-infrared distributed feedback quantum cascade laser both in the temporal and frequency domains. As opposed to the commonly observed route to chaos in semiconductor lasers, which involves undamping of the laser relaxation oscillations, quantum cascade lasers first exhibit regular self-pulsation at the external cavity frequency before entering into a chaotic low-frequency fluctuation regime. The bifurcation sequence, similar to that already observed in class A gas lasers under optical feedback, results from the fast carrier relaxation dynamics occurring in quantum cascade lasers, as confirmed by numerical simulations. Such chaotic behavior can impact various practical applications including spectroscopy, which requires stable single-mode operation. It also allows the development of novel mid-infrared high-power chaotic light sources, thus enabling secure free-space high bit-rate optical communications based on chaos synchronization.