[en] This thesis presents a measurement of the inelastic production of J/ψ mesons in ep collisions with the ZEUS detector at HERA using an integrated luminosity of 109 pb^-1. The J/ψ mesons were identified using the decay channel J/ψ → μ⁺μ^-. The measurements were performed in the kinematic range 2< Q²<80 GeV², 50< W<250 GeV, 0.2< z<0.9 and -1.6< Y_lab<1.3, where Q² is the virtuality of the exchanged photon, W is the photon-proton centre-of-mass energy, z is the fraction of the photon energy carried by the J/ψ meson in the proton rest frame and Y_lab is the rapidity of the J/ψ in the laboratory frame. The measured cross sections are compared to theoretical predictions within the non-relativistic QCD framework including colour-singlet and colour-octet contributions, as well as to predictions based on the k_T-factorisation approach. Calculations of the colour-singlet process generally agree with the data, whereas inclusion of colour-octet terms spoils this agreement. As a technical part of this thesis, the Straw-Tube Tracker (STT) GEANT simulation and track reconstruction software developed. Studies of the STT performance with MC data and real data presented. (orig.)