Bilayer graphene is normally a semimetal with parabolic dispersion, but a tunable bandgap up to few hundreds meV can be opened by breaking the symmetry between the layers through an external potential. Ab-initio calculations show that the optical response around the bandgap is strongly dominated by bound excitons, whose characteristics and selection rules differ from the usual excitons found in semiconductor quantum wells. In this work we study the physics of those excitons resonantly coupled to a photonic microcavity, assessing the possibility to reach the strong and the ultrastrong coupling regimes of light-matter interaction. We discover that both regimes are experimentally accessible, thus opening the way for a most promising technological platform, combining mid-infrared quantum polaritonics with the tunability and electronic features of graphene bilayers.