Excited states in the neutron-rich nucleus $^{68}\mathrm{Fe}$ were populated using a $^{9}\mathrm{Be}$($^{68}\mathrm{Co}, {}^{68}\mathrm{Fe}+\ensuremath{\gamma})\mathrm{X}$ charge-exchange reaction at 95 MeV/u. The new $\ensuremath{\gamma}$-ray transitions reported here for the first time complement data from $\ensuremath{\beta}$-decay studies and nucleon knockout reactions. In comparison to shell-model calculations with the LNPS effective interaction, two candidate states for the ${6}_{1}^{+}$ level emerge. The distinct population pattern of excited states and the magnitude of the cross section, ${\ensuremath{\sigma}}_{\mathrm{inc}}=0.51(6)$ mb, make this reaction a promising one for future in-beam $\ensuremath{\gamma}$-ray spectroscopy. Reaction calculations with nuclear structure input from a new, locally optimized Hamiltonian, $f7j4a$, together with general considerations for heavy-ion-induced charge-exchange reactions appear consistent with most of the observations, although challenges remain.