Résumé : An analysis of inelastic photoproduction of J/ψ mesons is presented using data collected at the ep collider HERA corresponding to an integrated luminosity of above 80 pb-1. Differential and double differential cross sections are measured in a wide kinematic region: 60 < Wγp < 260 GeV, 1 < pt,ψ2 < 60 GeV2 and 0.05 < z < 0.9, where z is the fraction of the energy of the exchanged photon transferred to the J/ψ meson in the rest frame of the target proton. Cross sections at z ≲ 0.3 are presented for the first time. Theoretical calculations within the Colour Singlet Model at NLO for direct photon processes are shown to give a good description of the data in the medium z region (0.3 < z < 0.9) up to the highest pt,ψ2 values. A calculation using a kt factorisation approach in LO in the Colour Singlet Model is also able to describe these data. The data in the full z range are also compared to LO calculations within a non-relativistic QCD framework including colour octet and colour singlet contributions for direct and resolved photons. It seems possible to reconcile data and theory with modest contributions from colour octet processes. The polarisation of the J/ψ meson is measured as a function of z and pt,ψ and is reasonably described by the theoretical predictions.