Circular dichroism spectroscopy is frequently used to characterize the chiral biomolecules by measuring the absorption spectra contrast between the left-handed circularly polarized light and the right-handed circularly polarized light. Compared with biomolecules, chiral metal plasmonic nanostructures also produce a strong circular dichroism response in the range of near-infrared. However, due to the large damping rate, the non-adjustable resonant frequency of the conventional metals, the applications of chiral metal plasmonic nanostructures in the fields of photoelectric detection and chemical and biochemical sensing are restricted. Here, we present a chiral graphene plasmonic Archimedes' spiral nanostructure that displays a significant circular dichroism response under the excitation of two polarizations of circularly polarized light. By manipulating the material and geometric parameters of the Archimedes' spiral, the stronger circular dichroism responses and modulation of the resonant wavelength are achieved. The optimized plasmonic nanostructure has outstanding refractive index sensing performance, where the sensitivity and figure of merit reach 7000nm/RIU and 68.75, respectively. Our proposed chiral graphene plasmonic Archimedes' spiral nanostructure might find potential applications in the fields of optical detection and high performance of index sensing.