Sequential double ionization and fragmentation dynamics of the H molecule exposed to an 750 nm, 4.5 fs elliptically polarized laser pulse is investigated by employing a quasi-classical model. In the model, momentum-dependent auxiliary potentials are added to the Hamiltonian to account for non-classical effects. Through theoretical exploitation of the molecular clock technique, the evolution of the vibrational wave packet of H formed by over-the-barrier ionization of the H molecule is tracked between the first and second ionization events with the temporal resolution of 140 attoseconds. The role of electron correlation in strong field ionization is captured. Our results show that the quasi-classical model is quite capable of describing and predicting light-induced multi-electron processes in the molecules. Our study provides a simple path of explaining and understanding the physical mechanism of the strong field multi-electron processes.