Random walk is one of the basic mechanisms found in many network applications. We study the epidemic spreading dynamics driven by biased random walks on complex networks. In our epidemic model, each time infected nodes constantly spread some infected packets by biased random walks to their neighbor nodes causing the infection of the susceptible nodes that receive the packets. An infected node get recovered from infection with a fixed probability. Simulation and analytical results on model and real-world networks show that the epidemic spreading becomes intense and wide with the increase of delivery capacity of infected nodes, average node degree, homogeneity of node degree distribution. Furthermore, there are corresponding optimal parameters such that the infected nodes have instantaneously the largest population, and the epidemic spreading process covers the largest part of a network.