We present new Hubble Space Telescope -Cosmic Origins Spectrograph (HST-COS) G130M spectroscopy which we have obtained for a sightline toward a filament projected 1.9 kpc from the nucleus of M87, near the edge of the inner radio lobe to the east of the nucleus. 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 ∼ 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 Lyα, C ii λ1335, and N v λ1238, finding v r = 147 ± 2 km s −1 , 138 ± 18 km s −1 , and 148 +14 −16 km s −1 relative to M87, and line broadenings σ r = 171 ± 2 km s −1 , 189 +12 −11 km s −1 , and 128 +23 −17 km s −1 respectively. We associate these three lines, as well as archival measurements of Hα, C iv λ1549, and He ii λ1640, 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] λ158µm flux observed by Herschel-PACS from this region implies the existence of a massive cold (T ∼ 10 3 K) component in the filament which contains significantly more mass (M ∼ 8000M within our r ≈ 100 pc sightline) than the FUV-emitting boundary layer. It has about the same bulk velocity and velocity dispersion as the boundary layer. We also detect [Fe xxi] λ1354 in emission at 4 − 5σ. This line is emitted from 1 keV (T ≈ 10 7 K) plasma, and we use it to measure the bulk radial velocity (v r = −92 +34 −22 km s −1 ) and velocity dispersion (σ r = 69 +79 −27 km s −1 ) of the plasma at this temperature. In contrast to the intermediate-temperature FUV lines, [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. A byproduct of our observations is the detection of absorption from interstellar gas in our Galaxy, observed in C ii λ1335 and Lyα.