We report measurements of the inclusive J/$\psi$ yield and average transverse momentum as a function of charged-particle pseudorapidity density ${\rm d}N_{\rm ch}/{\rm d}\eta$ in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with ALICE at the LHC. The observables are normalised to their corresponding averages in non-single diffractive events. An increase of the normalised J/$\psi$ yield with normalised ${\rm d}N_{\rm ch}/{\rm d}\eta$ is observed at mid-rapidity and backward rapidity, similar to that found earlier in pp collisions. At forward rapidity, a saturation of the relative yield is observed for high charged-particle multiplicities. The normalised average transverse momentum at forward and backward rapidity increases with multiplicity at low multiplicities and saturates beyond moderate multiplicities. In addition, the forward-to-backward nuclear modification factor ratio is also reported, showing an increasing suppression of J/$\psi$ production at forward rapidity with respect to backward rapidity for increasing charged-particle multiplicity.