We investigated a suspended bilayer graphene where the bottom/top layer is doped by boron/nitrogen substitutional atoms. By using density functional theory calculations, we found that at high dopant concentration (one B-N pair every 32 C atoms), the electronic structure of the bilayer does not depend on the B-N distance but on the relative occupation of the bilayer graphene sublattices by B and N. The presence of the dopants and the consequent charge transfer establish a built-in electric field between the layers, giving rise to an energy gap. We further investigated the electronic transport properties and found that intralayer current is weakly influenced by the presence of these dopants while the interlayer one is enhanced for biases, allowing an easy tunneling between layers. This effect leads to current rectification in asymmetric junctions.