We have studied the excitonic gap formation in the Bernal Stacked, bilayer graphene (BLG) structures at half-filling. Considering the local Coulomb interaction between the layers, we calculate the excitonic gap parameter and we discuss the role of the interlayer and intralayer Coulomb interactions and the interlayer hopping on the excitonic pair formation in the BLG. Particularly, we predict the origin of excitonic gap formation and condensation, in relation to the furthermost interband optical transition spectrum. The general diagram of excitonic phase transition is given, explaining different interlayer correlation regimes. The temperature dependence of the excitonic gap parameter is shown and the role of the chemical potential, in the BLG, is discussed in details.