  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 ${\rm d}N_{\rm ch}/{\rm d}\eta$, measured at
mid-rapidity, is observed at mid-rapidity and backward rapidity. 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 rapidities 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.
