  Using a weak-coupling renormalization group formalism, we study competing
ordered phases for repulsively interacting fermions on the bilayer honeycomb
lattice away from half-filling, which is realized experimentally as doped
bilayer graphene. As electrons are added to the system, excitonic order is
suppressed, and unconventional superconductivity appears generically in its
place. In general it is found that the maximum critical temperature for
superconductivity appears directly adjacent to the dome of particle-hole order,
illustrating the importance of fluctuations in these channels for the formation
of unconventional superconductivity. We obtain the phase diagram showing
characteristic ordering temperatures for both short- and long-ranged
interactions, and show that the most likely superconducting instabilities occur
in $d$-wave, $f$-wave, and pair density wave channels. The nature of and
competition between these phases are further analyzed using both free energy
expansion and self-consistent mean-field theory. The effects of finite
temperature and trigonal warping due to further-neighbor hopping are studied,
and implications for experiments on bilayer graphene are discussed.
