We investigate the detailed epidemic spreading process in scale-free networks with link weights that denote familiarity between two individuals. It is found that the spreading velocity reaches a peak quickly then decays in a power-law form. Numerical study exhibits that the nodes with larger strength is preferential to be infected, but the hierarchical dynamics are not clearly found, which is different from the well-known result in the unweighed network case. In addition, also by numerical study, we demonstrate that larger dispersion of weight of networks results in slower spreading, which indicates that epidemic spreads more quickly on unweighted scale-free networks than on weighted scale-free networks with the same condition.