The linear and weakly nonlinear transition regime of a hypersonic boundary layer on a sharp cone with circular cross section is investigated using Direct Numerical Simulations (DNS). In a natural transition scenario a broad disturbance spectrum is excited by free-stream disturbances leading to complex wave interactions. Therefore, in order to understand the natural transition process in hypersonic cone boundary layers, the flow was pulsed through a hole on the cone surface to generate a wavepacket with a wide range of disturbance waves. First, a two-dimensional DNS of a linear wavepacket was performed and results are compared to linear stability theory (LST). A good agreement was found to exist between DNS and LST results. We then performed DNS of a linear and a weakly nonlinear three-dimensional wavepacket. High frequency second-mode two-dimensional waves and low amplitude first-mode oblique waves were found to be present in the linear wavepacket. The high amplitude waves in the nonlinear wavepacket are modulated in the azimuthal direction due to weakly nonlinear interactions. A broad high amplitude frequency band was identified in the spectrum of the weakly nonlinear wavepacket, indicating fundamental resonance. Furthermore, secondary peaks were identified at approximately half the frequency of the high amplitude frequency band, which may be a consequence of subharmonic resonance.