Graphene terahertz (THz) surface plasmons provide hope for developing functional devices in the THz frequency. By coupling graphene surface plasmon polaritons (SPPs) and a planar waveguide (PWG) mode, Fano resonances are demonstrated to realize an ultrasensitive terahertz biosensor. By analyzing the dispersion relation of graphene SPPs and PWG, the tunable Fano resonances in the terahertz frequency are discussed. It is found that the asymmetric lineshape of Fano resonances can be manipulated by changing the Fermi level of graphene, and the influence of the thickness of coupling layer and air layer in sandwich structure on the Fano resonances is also discussed in detail. We then apply the proposed Fano resonance to realize the ultrasensitive terahertz biosensors, it is shown that the highest sensitivities of 3260 RIU-1 are realized. Our result is two orders of a conventional surface plasmon resonance sensor. Furthermore, we find that when sensing medium is in the vicinity of water in THz, the sensitivity increases with increasing refractive index of the sensing medium.