Infrared spectroscopy is the technique of choice for chemical identification of biomolecules through their vibrational fingerprints. However, infrared light interacts poorly with nanometric-size molecules. We exploit the unique electro-optical properties of graphene to demonstrate a high-sensitivity tunable plasmonic biosensor for chemically specific label-free detection of proteinmonolayers. The plasmon resonance of nanostructured graphene is dynamically tuned to selectively probe the protein at different frequencies and extract its complex refractive index. Additionally, the extreme spatial light confinement in graphene - up to two orders of magnitude higher than in metals - produces an unprecedentedly high overlap with nanometric biomolecules, enabling superior sensitivity in the detection of their refractive index and vibrational fingerprints. The combination of tunable spectral selectivity and enhanced sensitivity of graphene opens exciting prospects for biosensing.