Context. The Milky Way is surrounded by large amounts of diffuse gaseous matter that connects the stellar body of our Galaxy with its large-scale Local Group (LG) environment. Aims. To characterize the absorption properties of this circumgalactic medium (CGM) and its relation to the LG we present the so-far largest survey of metal absorption in Galactic high-velocity clouds (HVCs) using archival ultraviolet (UV) spectra of extragalactic background sources. The UV data are obtained with the Cosmic Origins Spectrograph (COS) onboard the Hubble Space Telescope (HST) and are supplemented by 21 cm radio observations of neutral hydrogen. Methods. Along 270 sightlines we measure metal absorption in the lines of Si ii, Si iii, C ii, and C iv and associated H i 21 cm emission in HVCs in the velocity range |vLSR| = 100 − 500 km s −1 . With this unprecedented large HVC sample we were able to improve the statistics on HVC covering fractions, ionization conditions, small-scale structure, CGM mass, and inflow rate. For the first time, we determine robustly the angular two point correlation function of the high-velocity absorbers, systematically analyze antipodal sightlines on the celestial sphere, and compare the HVC absorption characteristics with that of Damped Lyman α absorbers (DLAs) and constrained cosmological simulations of the LG (CLUES project). Results. The overall sky-covering fraction of high-velocity absorption is 77 ± 6 percent for the most sensitive ion in our survey, Si iii, and for column densities log N (Si iii) ≥ 12.1. This value is ∼ 4 − 5 times higher than the covering fraction of 21 cm neutral hydrogen emission at log N (H i) ≥ 18.7 along the same lines of sight, demonstrating that the Milky Way's CGM is multi-phase and predominantly ionized. The measured equivalent-width ratios of Si ii, Si iii, C ii, and C iv are inhomogeneously distributed on large and small angular scales, suggesting a complex spatial distribution of multi-phase gas that surrounds the neutral 21 cm HVCs. We estimate that the total mass and accretion rate of the neutral and ionized CGM traced by HVCs is MHVC ≥ 3.0×10 9 M⊙ and dMHVC/dt ≥ 6.1M⊙ yr −1 , where the Magellanic Stream (MS) contributes with more than 90 percent to this mass/mass-flow. If seen from an external vantage point, the Milky Way disk plus CGM would appear as a DLA that would exhibit for most viewing angles an extraordinary large velocity spread of ∆v ≈ 400 − 800 km s −1 , a result of the complex kinematics of the Milky Way CGM that is dominated by the presence of the MS. We detect a velocity dipole of high-velocity absorption at low/high galactic latitudes that we associate with LG gas that streams to the LG barycenter. This scenario is supported by the gas kinematics predicted from the LG simulations. Conclusions. Our study confirms previous results, indicating that the Milky Way CGM contains sufficient gaseous material to feed the Milky Way disk over the next Gyr at a rate of a few solar masses per year, if the CGM gas can actually reach the MW disk. We dem...