We present the results of nearly one year of gravity recording acquired at the active caldera of Campi Flegrei (CFc). CFc is one of the three active volcanoes in the Neapolitan area (southern Italy) and is currently the most active one. In fact, the CFc is undergoing a period of unrest characterised by slow uplift of the ground, a peculiar phenomenon known worldwide as bradyseism, accompanied by seismicity and intense fumarolic emissions. Due to the increased intensity of the volcano dynamics, a permanent gravity station equipped with a gPhoneX spring gravimeter was installed to enhance the geodetic monitoring programmes. The purpose of the continuous recordings is to complement the time-lapse observations carried out periodically on networks of benchmarks, in order to continuously monitor the short-term gravity signals. We report on the various processing steps and analyses performed to obtain reliable parameters of the Earth's tides, non-tidal corrections and gravity residuals. The various methodologies employed to investigate the instrumental drift are also elucidated in depth, because it may masquerade the elusive gravity changes resulting from mass fluctuations within the volcanic and geothermal systems. Residual gravity signals, retrieved from the recordings, after reduction of body and ocean tides, polynomial drift, atmospheric, tilt and change of the Earth Orientation Parameters (EOP) effects, appear to be uncorrelated with hydrology, while they show a clear correlation with the most energetic earthquakes, that strongly characterise the current bradyseismic crisis. The residual gravity signals display peculiar trends characterized by steps or offsets (up to about 600 nm/s2) and transients in coincidence of the most energetic volcano-tectonic events (Magnitude > 2.5) and seismic swarms. The steps in the gravity residuals are likely to be of instrumental origin, while the amplitudes of the observed transients are not consistent with co-seismic or volcanological phenomena, for which there is no evidence from other monitoring techniques. Unfortunately, the lack of repeated absolute gravity measurements severely limits our ability to attribute the observed gravity variations to geological sources. From the analysis of the gravity records, reliable tidal gravity models have been derived, which will improve the accuracy of volcano monitoring by allowing a precise reduction of tidal effects for both relative and absolute gravity measurements taken in these volcanic areas.