  We present new HST-COS G130M spectroscopy for a sightline toward a filament
projected 1.9 kpc from the nucleus of M87. The combination of the sensitivity
of COS and the proximity of M87 allows us to study the structure of this
filament in unparalleled detail. We propose that the filament is composed of
many cold clumps, each surrounded by an FUV-emitting boundary layer, with the
filament having a radius $r_c \sim 10$ pc and the clumps filling the cylinder
with a low volume filling factor. The observed velocity dispersion in emission
lines from the filament results from the random motions of these clumps within
the filament.
  We measure fluxes and kinematics for emission lines of Ly$\alpha$, C II
$\lambda$1335, and N V $\lambda1238$. We associate these three lines, as well
as archival measurements of H$\alpha$, C IV $\lambda$1549, and He II
$\lambda$1640, with a multitemperature boundary layer around clumps which are
moving with supersonic random motions in the filament. This boundary layer is a
significant coolant of the hot gas. We show that the [C II] $\lambda$158$\mu$m
flux observed by Herschel-PACS from this region implies the existence of a
massive cold ($T \sim 10^3$ K) component in the filament which contains
significantly more mass than the FUV-emitting boundary layer.
  We also detect [Fe XXI] $\lambda$1354 in emission at $4-5\sigma$. This line
is emitted from 1 keV ($T \approx 10^7$ K) plasma, and we use it to measure the
bulk radial velocity and velocity dispersion of the plasma at this temperature.
In contrast to the intermediate-temperature FUV lines, [\ion{Fe}{xxi}] is
blueshifted relative to M87 and matches the bulk velocity of a nearby filament
to the south. We hypothesize that this line arises from the approaching face of
the radio bubble expanding through this sightline, while the filament lies on
the receding side of the bubble. (abstract is abridged)
