In the present study, an optical fiber surface plasmon resonance (SPR) biosensor was developed for measuring time- and concentration-dependent DNA hybridization kinetics. Its design complies with a 3D $\mathrm{Au}/{\mathrm{Al}}_2{\mathrm{O}}_3$ multilayer composite hyperbolic metamaterial (HMM), a graphene film, and a D-shaped plastic optical fiber. According to the numerical simulation and the experimental demonstration, the SPR peak of the designed biosensor can be effectively altered in the range of visible to near-infrared by varying the HMM structure. The sensitivity of the appliance was shown to achieve a value of up to 4461 nm/RIU, allowing its applicability for bulk refractive index sensing. Furthermore, a biosensor designed in this work displayed high-resolution capability (ranging from 10 pM to 100 nM), good linearity, and high repeatability along with a detection limit down to 10 pM, thus suggesting a vast potential for medical diagnostics and clinical applications.