The production of charm jets in proton-proton collisions at a center-of-mass energy of $\sqrt{s}=7$ TeV was measured with the ALICE detector at the CERN Large Hadron Collider. The measurement is based on a data sample corresponding to a total integrated luminosity of $6.23$ ${\rm nb}^{-1}$, collected using a minimum-bias trigger. Charm jets are identified by the presence of a D$^0$ meson among their constituents. The D$^0$ mesons are reconstructed from their hadronic decay D$^0\rightarrow$K$^{-}\pi^{+}$. The D$^0$-meson tagged jets are reconstructed using tracks of charged particles (track-based jets) with the anti-$k_{\mathrm{T}}$ algorithm in the jet transverse momentum range $5<~p_{\rm{T,jet}}^{\mathrm{ch}}<~30$ ${\rm GeV/}c$ and pseudorapidity $|\eta_{\rm jet}|<~0.5$. The fraction of charged jets containing a D$^0$-meson increases with $p_{\rm{T,jet}}^{\rm{ch}}$ from $0.042 \pm 0.004\, \mathrm{(stat)} \pm 0.006\, \mathrm{(syst)}$ to $0.080 \pm 0.009\, \rm{(stat)} \pm 0.008\, \rm{(syst)}$. The distribution of D$^0$-meson tagged jets as a function of the jet momentum fraction carried by the D$^0$ meson in the direction of the jet axis ($z_{||}^{\mathrm{ch}}$) is reported for two ranges of jet transverse momenta, $5<~p_{\rm{T,jet}}^{\rm{ch}}<~15$ ${\rm GeV/}c$ and $15<~p_{\rm{T,jet}}^{\rm{ch}}<~30$ ${\rm GeV/}c$ in the intervals $0.2<~z_{||}^{\rm{ch}}<~1.0$ and $0.4<~z_{||}^{\rm{ch}}<~1.0$, respectively. The data are compared with results from Monte Carlo event generators (PYTHIA 6, PYTHIA 8 and Herwig 7) and with a Next-to-Leading-Order perturbative Quantum Chromodynamics calculation, obtained with the POWHEG method and interfaced with PYTHIA 6 for the generation of the parton shower, fragmentation, hadronisation and underlying event.