  In the framework of the $k_T$-factorization approach, we analyse the
inclusive and inelastic production of $J/\psi$ particles in deep inelastic $ep$
scattering. We take into account both colour-singlet and colour-octet
production channels. We inspect the sensitivity of theoretical predictions to
the choice of model parameters. Our theoretical results agree reasonably well
with recent experimental data collected by the collaboration H1 at HERA.
