Maybe someone thought of and posted posted this already, but I'm going crazy with it so here goes. It now seems so obvious that I know someone must have thought of it before, but I already wrote it all and didn't take the time to search for this theory, so read on. Try the TL;DR.
We want the Rune of the Darkening, or so I am told. Shadowmourne, the sister blade of Frostmourne, is something of a dark Frostmourne. Like a shadowy Frostmourne, if you will. So, to darken the reforged Frostmourne (and, as it is reforged, should not it receive infusion once more?), might we infuse it the same as we did Shadow's Edge? I dunno how they forged Frost, I don't read books, but Shadow's Edge, once forged, was infused to become Shadomourn by players thusly:
  1. Farm souls. Many souls, to be precise.
  2. Turn into abomination. Abomination eats green goo.
  3. Blood queen infection.
  4. Frost dragon blowing us.
  5. Kills bosses for shards of the Frozen Throne crystal.
Now, in Legion, we got the Violet Hold instance again, and lo and behold those exact three things are within: Festerface, Blood-Princess Thal'ena and Shivermaw, respectively. Not only are they present, they have incredibly similar mechanics to the ones necessary for the Shadowmourne infusion quests:
  1. Festerface with vom up some goo and eat it
  2. Blood Princess Thal'ena will infect players
  3. Shivermaw has frost breath
Maybe Blizzard is being lazy with recurring mechanics, and these may just be offhand references to the fact that Dalaran was previously in Northrend and therefore all its prisoners are from Wrath, but these three are so specific that it gnaws at me. If you're looking to infuse a -mourne in a dark way, these three seem to be the way to do it, and we got 'em here in the Broken Isles.
[As previously posited by others, Death Knights and Paladins have an adorable mirroring in WoW, and as Ret has a long quest chain to swap their Ashbringer to the Corrupted Ashbringer once more, following an old theorized quest chain to change their previously-impossible-to-obtain-legendary (Ashbringer) into the epic which was obtainable (Corrupted Ashbringer), perhaps Death Knights use an old questline to change their previously-impossible-to-obtain-legendary Frostmourne(s) into a version reminiscent of their possible-to-obtain Shadowmourne (no, the skin doesn't remind me of Shadowmourne either, but the Darkening bit does, so I'm running with it). I don't think it's a stretch, but it's also 2:30 in the morning, so I might not be the best judge.]

My theory is based on the old Shadow's Edge infusion quest line:
First, you gotta farm souls. Shadomorn needed souls from ICC specifically, though lorewise this was a non-sequitur and was just to gate off farming kobold souls, but maybe we do need souls from a specific undead-filled place, or specifically undead souls. I'm guessing not, but I posit it nonetheless. Shadmo also just needed to apply a debuff to enemies to take their souls by hitting them once, so I assume this is similar in simply needing to hit something once for it to count. In any case, we farm souls with Blades of the Fallen Prince first, cause Mournes want souls first. This will unlock the next three phases involving Assault on Violet Hold instance. Maybe I'm missing an initial step, though I believe, if we are to mirror the Shamo forging and infusion, that reforging Frostmourne into the artifact we acquire was the initial quest, followed by this second quest of soul reaping.
So, if correct, it follows that the next step in unholy infusion, with the vomiting abomination and such. These next three might be doable in any order, or a new specific order, but I'm gonna list them in the mirrored order. SM needed you to turn into an abomination, which I can't seem to do. I can summon one if I'm unholy but that doesn't help me infuse my BotFP. Here's a quote from the now-disarmed Mograine regarding unholy vomit:
Find a way to absorb his mutated slime. Use the foul substance to infuse your weapon with the power of the unholy.
I'm not the kind of DK to blow off doing instance mechanics properly, so I dunno if there's a debuff if you stand in slime, which if there is, is definitely my thought here, but I'm guessing that just standing in the green piles of vom for some time will trigger this part. I believe standing in and concurrently killing the goo makes it apply a debuff, so maybe that. SM needed 100 energy from 3 piles of goo, so I'm gonna say 3 piles worth of goo debuff or whatever - this is also the amount of goo per vom spray in normal AoVH. Maybe you have to slay the special black bile with BotFP, maybe you have to be hit by the vom spray, maybe you have to be so sick of searching for this skin that you hurl on your keyboard. But Festerface gives the shadow-unholy infusion. Obviously, you then will then need to kill Festerface.
Next up is Blood Princess Thal'ena. SM needed to do a somewhat complex buff trading sequence using Blood Mirror from the Blood Queen, which led to a quester-specific debuff which then necessitated gaining the other debuff from the Blood Queen, Essence of the Blood Queen. Only the final most debuff is relevant here, as we can nowadays obtain Essence of the Blood Princess. I'm not entirely familiar with this part, so here's Wowhead's take:
The top DPS'er gets her Essence of the Blood Queen debuff (also known as her "bite"). Once bitten the player waits the full sixty seconds for the debuff to run out and convert to Frenzied Bloodthirst, then uses Vampiric Bite on another player. Biting another player increases the number of stacks of Gushing Wound the player has, increasing the damage the debuff does and getting the player closer to completing their quest. The player must use bite three times, and then defeat Blood-Queen Lana'thel in order to complete the quest.
So, my theory for this one is: Essence of the Blood Princess has its own bite mechanic in the same fashion as the Queen's when it wears off, so we perfectly mirror the old way here and simply get the essence debuff, let it wear off and then bite some poor teammate, repeating this thrice. Now, if these are hidden achievement trackers, this is where no one has progressed in the hidden skin questline, as no one gets the debuff thrice in the same fight, because no one is that slow to DPS down this boss, not even 7.0 frost DKs. This will take a coordinated effort, as it did previously for SM, even after WotLK. That's shadow-blood infusion, bite your teammates three times. And then kill the princess, obviously.
Lastly, Shivermaw. She drops a sweet mandible, and believe it or not, the undead dragon has frost breath. SM needed four gouts of Sindragosa's breath, but I guess Frostmourne is frosty enough so Shivermaw will do for us now. Easy peasy, get hit by Shivermaw's Frost Breath four times. I don't think you need to be hit by ice bomb if we're mirroring the old quests, but maybe you need a stronger ice attack because Shivermaw's assuredly so much weaker than Sindragosa. Either way, my final guess here is get hit by Shivermaw's breath four times, then kill Shivermaw. That's shadow-frost infusion.
Now, the final bit I'm stuck on. I haven't tested any of these theories yet, but if there is a fourth part, which there should be since Runes of the Darkening don't just pop up in your inventory for being hit by boss mechanics, it will mirror collecting the shards of Frozen Throne crystal. My guess is it will probably be a scenario or instance given by Highlord Darion Mograine (lead designer of Shadowmourne) once we infuse BotFP with the three thingies. Lore-wise, the shards are the machination of Kil'jaeden to hold Ner'Zhul in the Frozen Throne, so if you wanna brainstorm, think fel crystals or Lich King prisons.

TL;DR We darken Frostmourne the same way we darkened Shadow's Edge into Shadowmourne using the same quest mechanics, which are readily available in the Violet Hold instance thusly:
(Get 1000 souls? 50 souls? No souls? Souls = kills?)
Stand in Festerface's goo pile as it dies to get goo debuff, thrice(?) in the same fight, kill Festerface;
Get debuffed by Blood Princess Thal'ena's "Essence of the Blood Princess," let it wear off, then bite a teammate in the time before you get charmed, doing this thrice in the same fight, kill Blood Princess Thal'ena;
Get hit by Shivermaw's Frost Breath four times in the same fight, kill Shivermaw;
???
Rune get

Obviously, not all in the same run of AoVH, and while having Blades of the Fallen Prince equipped.

I really hope no one's tried this yet. It is now 4am. I will be attempting these tactics when I wake later today.