A small-scale, two-phase (liquid/gas) xenon time projection chamber (Xurich II) was designed, constructed and is under operation at the University of Zürich. Its main purpose is to investigate the microphysics of particle interactions in liquid xenon at energies below 50 keV, which are relevant for rare event searches using xenon as target material. Here we describe in detail the detector, its associated infrastructure, and the signal identification algorithm developed for processing and analysing the data. We present the first characterisation of the new instrument with calibration data from an internal Kr source. The zero-field light yield is 15.0 and 14.0 photoelectrons/keV at 9.4 and 32.1 keV, respectively, and the corresponding values at an electron drift field of 1 kV/cm are 10.8 and 7.9 photoelectrons/keV. The charge yields at these energies are 28 and 31 electrons/keV, with the proportional scintillation yield of 24 photoelectrons per one electron extracted into the gas phase, and an electron lifetime of 200 s. The relative energy resolution, , is 11.9 and 5.8% at 9.4 and 32.1 keV, respectively using a linear combination of the scintillation and ionisation signals. We conclude with measurements of the electron drift velocity at various electric fields, and compare these to literature values.