Understanding the peculiar properties of ultra-diffuse galaxies (UDGs) via spectroscopic analysis is a challenging task requiring very deep observations and exquisite data reduction. In this work, we perform one of the most complete characterizations of the stellar component of UDGs to date using deep optical spectroscopic data from OSIRIS at Gran Telescopio CANARIAS (GTC). We measure radial and rotation velocities, star formation histories (SFHs), and mean population parameters, such as ages and metallicities, for a sample of five UDG candidates in the Coma cluster. From the radial velocities, we confirm the Coma membership of these galaxies. We find that their rotation properties, if detected at all, are compatible with dwarf-like galaxies. The SFHs of the UDG are dominated by old (˜7 Gyr), metal-poor ([M/H] ˜ -1.1),and α-enhanced ([Mg/Fe] ˜ 0.4) populations followed by a smooth or episodic decline that halted ˜2 Gyr ago, possibly a sign of cluster-induced quenching. We find no obvious correlation between individual SFH shapes and any UDG morphological properties. The recovered stellar properties for UDGs are similar to those found for DDO 44, a local UDG analogue resolved into stars. We conclude that the UDGs in our sample are extended dwarfs whose properties are likely the outcome of both internal processes, such as bursty SFHs and/or high-spin haloes, as well as environmental effects within the Coma cluster.