We present a census of neutral gas in the Milky Way disk and halo down to limiting column densities of $N$(HI)$sim10^{14}$ cm$^{-2}$ using measurements of HI Lyman-series absorption from the Far Ultraviolet Spectroscopic Explorer (FUSE). Our results are drawn from an analysis of 25 AGN sightlines spread evenly across the sky with Galactic latitude |b|$gtrsim 20^{circ}$. By simultaneously fitting multi-component Voigt profiles to 11 Lyman-series absorption transitions covered by FUSE (Ly$beta$-Ly$mu$) plus HST measurements of Ly$alpha$, we derive the kinematics and column densities of a sample of 152 HI absorption components. While saturation prevents accurate measurements of many components with column densities 17$lesssim$log$N$(HI)$lesssim$19, we derive robust measurements at log$N$(HI)$lesssim$17 and log$N$(HI)$gtrsim$19. We derive the first ultraviolet HI column density distribution function (CDDF) of the Milky Way, both globally and for low-velocity (ISM), intermediate-velocity clouds (IVCs), and high-velocity clouds (HVCs). We find that IVCs and HVCs show statistically indistinguishable CDDF slopes, with $beta_{rm IVC}=$ $-1.01_{-0.14}^{+0.15}$ and $beta_{rm HVC}=$ $-1.05_{-0.06}^{+0.07}$. Overall, the CDDF of the Galactic disk and halo appears shallower than that found by comparable extragalactic surveys, suggesting a relative abundance of high-column density gas in the Galactic halo. We derive the sky covering fractions as a function of HI column density, finding an enhancement of IVC gas in the northern hemisphere compared to the south. We also find evidence for an excess of inflowing HI over outflowing HI, with $-$0.88$pm$0.40 M$_odot$ yr$^{-1}$ of HVC inflow versus 0.20$pm$0.10 M$_odot$ yr$^{-1}$ of HVC outflow, confirming an excess of inflowing HVCs seen in UV metal lines.
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