There is a Death Knight in my campaign that, eventually, the PCs will likely want to face and destroy. I haven't fully fleshed out his lore yet as the PCs have been focused elsewhere and haven't tried to find much out about him, but this is about to change.

I need some ideas of what conditions might need to be met to permanently destroy this particular Death Knight (per MM, Death Knights return in a day or two if killed by conventional means, each has some specific way it can die/be killed permanently).

Alternatively the PCs could end up imprisoning him, but they like to kill things (shocker), so would be good to have a way they could - would motivate them.

What I do know about the Death Knight - 

I have two Goddesses that rule Feywild, Maab (Winter theme), and Titania (Summer). In the distant past the goddesses were friendly, today they are hostile. Same applies to their priesthoods. In life the Death Knight was a Paladin devoted to Maab. That was long ago when the goddesses were still friendly. Whatever he did that turned him into a Death Knight may be involved in why the goddesses are hostile towards each other now. He has been imprisoned for a thousand years or more in an abandoned temple devoted to both goddesses. The PCs released him (by accident) a while back.

He has reclaimed a ruined tower that was once his and turned the surrounding forest dark and evil. This last fact has come to the attention of the ruler of the land, and the PCs. Oh! and one of my PCs is a Paladin devoted to Maab himself, who has also muti-classed into Warlock with Titania as his Patron, when he needed a miracle to resurrect his brother (another PC) that was slain by said Death Knight when the PCs released him.

 That's pretty much Tolkien 101.

Death Knights can be slain by the mystic blade of ________, it was fashioned by the Wizard _________, but he was slain by the Winter Fey when they learned he was creating it, and the blade spirited off into alternate plane of _____________. The blade is a one-shot weapon; it will violently destroy itself and the Death Knight's soul if the final killing stroke is made with this weapon. Done - one Death Knight dispatched, and the Party doesn't get to run around with a permanent Death Knight killer. The Winter Fey didn't/couldn't actually destroy the weapon as the only thing that can do that is allow the enchantment and the soul of a Death Knight to annihilate one another. 

The party can quest for it: steal it, bargain for it, fight their way into the Fey sanctuary of _______________, etc. to gain the blade.

I take it, based on this thread, that you're still planning on shoving your Party down this plotline, even though they seem to not want to go there.

Thanks for the suggestion, nice and flexible, appreciated. 

As to the question, there was never any plan to railroad them to do anything. They are level 10, big stuff is happening in the world. They have made connections with powerful people and are in the know about major events and happenings.

They have been doing what they want, choosing their own path since level 3. The world is big and much more is going on than they can address personally. They just wrapped up an arc that took a year in game and out, I am giving them updates on what has happened while they have been away. They can pursue any or NONE of them. I remind them of this often, they decide what they do and the story is about them.

If they decide to never go near the death knight again, that's great. The Death Knight is doing his thing and will continue to if left alone, just like all the other NPCs in the game. I think there's a lot of fun for the group to be had with him in time (one of my biggest fears was that they would charge in at him, vastly unprepared), and my plot lines interweave. So even though they may not pursue him now or EVER, I'm at a place where knowing more about his lore is useful, as he will be interacting with other NPCs the PCs know. Hence my question.

As proof of my good intent towards my players: in last night's session one PCs decided to meet with one of the Queen's advisors before the next council. I had the advisor invite the other PCs. They all decided to go, asked lots of questions, and decided one of the other threats was more urgent (raids in the north, possibly as insurgence). Believing themselves more capable than the army (they are probably not wrong) they want to face that threat themselves. They convinced the advisor to agree with their plan, and with that combined support, the Queen took their advice.

One of the PC's rivals, a local noble has been tasked with relocating villagers near the Death Knight's forest, blocking nearby roads, and observing.

Of course, this is all back burner now, as it looks like the Winter Fey Paladin PC just volunteered the group to help attack the High Priestest of the Summer Fey Temple, in retaliation for the Summer Fey poisoning to death the Winter Fey high priestess a while back (using a poisoned apple the same PC gave to the summer Fey as part of their warlock RP).

I had thought to offer up to the group the conflict between the Faith's as a possible arc to heal the divide between the goddesses, but depending on how they play this out, its looking more likely to go very much the opposite! Entertaining either way.

My DM threw a Death Knight at a level 5 party to use as a threat to avoid, never intending for us to kill him, but once we discovered that he used to be an honorable paladin and was actually cursed to undeath, our focus shifted entirely from destroy to redeem. With a lot of help from NPC clerics and a masterful (30+) Persuasion roll from my bard, we managed to talk him into repentance and back into the good graces of Lathander Morninglord. It was a coup that my DM did not expect at all. So there's that possibility.