The electron transport properties of the system consisting of the zigzag graphene nanoflake doped with nitrogen and boron atoms connected to two Au electrodes through S-Au bonds are investigated theoretically. The results show that a nanoflake doped with nitrogen and boron atoms at edges has poor rectifying performance. While the system consisting of two pieces of graphene flakes doped by boron and nitrogen atoms, respectively, and linked with an alkane chain, shows good performance. And the significant effects of the doped sites on the current-voltage characteristics are observed. The mechanisms for these phenomena are explained by the different shifts of transmission spectra, the different spatial distributions of the molecular projected self-consistent Hamiltonian eigenstates. The negative differential resistance behavior results from the biase induced shifts of the energy level and change of the resonance transmission spectra, and the suppression of the relevant channels at some bias voltages.