Three-dimensional (3D) bioprinting allows the production of artificial 3D cellular microenvironments thanks to the controlled spatial deposition of bioinks. Proper bioink characterization is required to achieve the essential characteristics of printability and biocompatibility for 3D bioprinting. In this work, a protocol to standardize the experimental characterization of a new bioink is proposed. A functionalized hydrogel based on gelatin and chitosan was used. The protocol was divided into three steps: pre-printing, 3D bioprinting, and post-printing. For the pre-printing step, the hydrogel formulation and its repeatability were evaluated. For the 3D-bioprinting step, the hydrogel-printability performance was assessed through qualitative and quantitative tests. Finally, for the post-printing step, the hydrogel biocompatibility was investigated using UMR-106 cells. The hydrogel was suitable for printing grids with good resolution from 4 h after the cross-linker addition. To guarantee a constant printing pressure, it was necessary to set the extruder to 37 °C. Furthermore, the hydrogel was shown to be a valid biomaterial for the UMR-106 cells’ growth. However, fragmentation of the constructs appeared after 14 days, probably due to the negative osteosarcoma-cell interference. The protocol that we describe here denotes a strong approach to bioink characterization to improve standardization for future biomaterial screening and development.