We present an analytical device model for a graphene bilayer field-effect transistor (GBL-FET) with a graphene bilayer as a channel and with back and top gates. The model accounts for the dependences of the electron and hole Fermi energies as well as energy gap in different sections of the channel on the bias back-gate and top-gate voltages. Using this model, we calculate the dc and ac source-drain currents and the transconductance of GBL-FETs with both ballistic and collision dominated electron transport as functions of structural parameters, the bias back-gate and top-gate voltages, and the signal frequency. It is shown that there are two threshold voltages, Vth,1 and V th,2, so that the dc current versus the top-gate voltage relation markedly changes depending on whether the section of the channel beneath the top gate (gated section) is filled with electrons, depleted, or filled with holes. The electron scattering leads to a decrease in the dc and ac currents and transconductances, whereas it weakly affects the threshold frequency. As demonstrated, the transient recharging of the gated section by holes can pronouncedly influence the ac transconductance resulting in its nonmonotonic frequency dependence with a maximum at fairly high frequencies.