We observed GRB 190114C (redshift z = 0.4245), the first gamma-ray burst (GRB) ever detected at TeV energies, at optical and near-infrared wavelengths with several ground-based telescopes and the Hubble Space Telescope, with the primary goal of studying its underlying supernova, SN 2019jrj. The monitoring spanned the time interval between 1.3 and 370 days after the burst, in the observer frame. We find that the afterglow emission can be modelled with a forward shock propagating in a uniform medium modified by time-variable extinction along the line of sight. A jet break could be present after 7 rest-frame days, and accordingly the maximum luminosity of the underlying supernova (SN) ranges between that of stripped-envelope core-collapse SNe of intermediate luminosity and that of the luminous GRB-associated SN 2013dx. The observed spectral absorption lines of SN 2019jrj are not as broad as in classical GRB SNe and are instead more similar to those of less-luminous core-collapse SNe. Taking the broad-lined stripped-envelope core-collapse SN 2004aw as an analogue, we tentatively derive the basic physical properties of SN 2019jrj. We discuss the possibility that a fraction of the TeV emission of this source might have had a hadronic origin and estimate the expected high-energy neutrino detection level with IceCube.