The production of J/ψ and ψ(2S) was studied with the ALICE detector in Pb-Pb collisions at the LHC. The measurement was performed at forward rapidity (2.5 < y < 4) down to zero transverse momentum (pt) in the dimuon decay channel. Inclusive J/ψ yields were extracted in different centrality classes and the centrality dependence of the average pt is presented. The J/ψ suppression, quantified with the nuclear modification factor (RAA), was measured as a function of centrality, transverse momentum and rapidity. Comparisons with similar measurements at lower collision energy and theoretical models indicate that the J/ψ production is the result of an interplay between color screening and recombination mechanisms in a deconfined partonic medium, or at its hadronization. Results on the ψ(2S) suppression are provided via the ratio of ψ(2S) over J/ψ measured in pp and Pb-Pb collisions.