This paper investigates the global dynamics and bifurcation structure of a viral infection logistic model with delayed nonlinear CTL response and periodic immune response. It is proved that the basic reproduction numbers, R0 and R1, determine the outcome of viral infection. Besides changes in the amplititude of lytic component, we show, via numerical simulations, that , the birth rate of susceptible host cells and the maximum proliferation of target cells are crucial to the outcome of a viral infection. Time delay can alter the period of oscillation for the larger level of periodic forcing. Period doubling bifurcations of the system are observed via simulations. Our results can provide a possible explanation of the oscillation behaviors of virus population,which were observed in chronic HBV or HCV carriers.