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Localized surface plasmon resonance (LSPR) is a promising detection method for label-free sensing of biomolecules. In this paper, a multilayer design for a LSPR biosensor is presented. In the proposed design, a periodic array of dielectric grating is incorporated on top of a graphene layer in the biosensor. The aim is to improve sensitivity of the LSPR biosensor through monitoring biomolecular interactions of biotin-streptavidin. Sensitivity improvement is obtained for the proposed LSPR biosensor compared with conventional SPR counterparts. In addition, to optimize the design, we have investigated grating geometry including volume factor and grating depth. The outcome of this investigation identifies ideal functioning conditions corresponding to the best design parameters.