With the aim of better understanding recent experimental results, we performed density functional theory calculations (DFT), including van der Waals interactions, on bilayer graphene over a Ru(0001) surface. Two stacking sequences (AB and AA) for bilayer graphene were considered and compared with monolayer graphene. For each case relaxed atomic positions, calculated STM images and density of states were obtained and these are discussed in detail. Our results suggest that moiré patterns of graphene over a Ru(0001) surface have a remarkable electronic influence, whose origin is the coupling of graphene layers and the Ru(0001) surface. Additionally, we found that atomic lattice observed by STM on such moiré patterns is related with stacking sequence of bilayer graphene.