We report measurements of the inclusive J/ψ yield and average transverse momentum as a function of charged-particle pseudorapidity density $d{N}_{\mathrm{ch}}/d\eta$ in p–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02\text{TeV}$ with ALICE at the LHC. The observables are normalised to their corresponding averages in non-single diffractive events. An increase of the normalised J/ψ yield with normalised $d{N}_{\mathrm{ch}}/d\eta$, 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/ψ production at forward rapidity with respect to backward rapidity for increasing charged-particle multiplicity.