We study the halo mass accretion history (MAH) and its correlation with the internal structural properties in coupled dark energy cosmologies (cDE). To accurately predict all the non-linear effects caused by dark interactions, we use the COupled Dark Energy Cosmological Simulations (CoDECS). We measure the halo concentration at z=0 and the number of substructures above a mass resolution threshold for each halo. Tracing the halo merging history trees back in time, following the mass of the main halo, we develope a MAH model that accurately reproduces the halo growth in term of M_{200} in the {\Lambda}CDM Universe; we then compare the MAH in different cosmological scenarios. For cDE models with a weak constant coupling, our MAH model can reproduce the simulation results, within 10% of accuracy, by suitably rescaling the normalization of the linear matter power spectrum at z=0, {\sigma}_8. However, this is not the case for more complex scenarios, like the "bouncing" cDE model, for which the numerical analysis shows a rapid growth of haloes at high redshifts, that cannot be reproduced by simply rescaling the value of {\sigma}_8. Moreover, at fixed value of {\sigma}_8, cold dark matter (CDM) haloes in these cDE scenarios tend to be more concentrated and have a larger amount of substructures with respect to {\Lambda}CDM predictions. Finally, we present an accurate model that relates the halo concentration to the time at which it assembles half or 4% of its mass. Combining this with our MAH model, we show how halo concentrations change while varying only {\sigma}_8 in a {\Lambda}CDM Universe, at fixed halo mass.