X-ray computed tomography (CT) is an important non-destructive imaging technique, particularly in clinical diagnostics. Even with the latest innovations like dual-energy and photon-counting CT, the image contrast is solely generated from attenuation in the tissue. An extension – fully compatible with these novelties – is dark-field CT, which retrieves an additional, so-called dark-field contrast. Unlike the attenuation channel, the dark-field channel is sensitive to tissue microstructure and porosity below the resolution of the imaging system, which allows additional insights into the health of the lung tissue or the structure of calcifications. The potential clinical value has been demonstrated in several preclinical studies and recently also in radiography patient studies. Just recently the first dark-field CT for the human body was established at the Technical University of Munich and in this paper, we discuss the performance of this prototype. We evaluate the interferometer components and the imposed challenges that the integration into the CT gantry brings by comparing the results to simulations and measurements at a laboratory setup. The influence of the clinical X-ray source on the Talbot-Lau interferometer and the impact of vibrations, which are immanent on the clinical CT gantry, are analyzed in detail to reveal their characteristic frequencies and origin. A beam hardening correction is introduced as an important step to adapt to the poly-chromatic spectrum and make quantitative dark-field imaging possible. We close with an analysis of the image resolution and the applied patient dose, and conclude that the performance is sufficient to suggest initial patient studies using the presented dark-field CT system.