  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.
