Bilayer graphene is a highly tunable material: not only can one tune the Fermi energy using standard gates, as in single-layer graphene, but the band structure can also be modified by external perturbations such as transverse electric fields or strain. We review the theoretical basics of the band structure of bilayer graphene and study the evolution of the band structure under the influence of these two external parameters. We highlight their key role concerning the ease to experimentally probe the presence of a Lifshitz transition, which consists in a change of Fermi contour topology as a function of energy close to the edges of the conduction and valence bands. Using a device geometry that allows the application of exceptionally high displacement fields, we then illustrate in detail the way to probe the topology changes experimentally using quantum Hall effect measurements in a gapped bilayer graphene system.