kicking ass and taking names
These are some basic types of Hounding Foes I use in my games. Keep in mind there is considerable overlap (Michael Myers is Relentless, Invincible and Faceless.
The Relentless Opponent
The Relentless Opponent can be an evil villain like Vader, but doesn't have to be. Examples of good relentless foe are Brienne of Tarth and the Archangel Michael. They are usually solitary, though can be servants of a greater power or threat. The distinguishing feature here is determination and constant effort. When you throw a relentless foe at the party they should feel she might attack any moment, and her appearances ought to be fairly regular. The Relentless Opponent wants nothing more than to get the party and either capture them or kill them.
The Invincible Foe
The Invincible Foe may be some form of undead, or some kind of construct (Vader or The Terminator). They can also simply just be tough. The Invincible Foe can be destroyed but it takes a lot of effort, and there are often many ways for him to escape death. My favorite invincible foes in D&D are golems (particularly Flesh Golems). I think they make a great, heavy-duty threat that keeps most parties busy. I like to give my Invincible Foes a weakness of some kind, something the party can learn about through effort and use against the creature. Sometimes the party thinks they've killed an Invincible Foe only to encounter them a week later after they've regenerated.
The Faceless Evil
RPGs are not a visual medium, but still something about an NPC in a mask seems to have an impact on peoples' imaginations. I've used a lot of masked opponents in my games. Lord Soth is a good example of a Faceless Evil. Something about the bucket helmet makes him more menacing.
I think part of the intrigue with the Faceless Evil is the riddle it suggests about their past. The players want to know why this person (if it is a person) covers her face. Sometimes it is just a mater of disfiguration like with Phantom of the Opera, other times it is a much more involved backstory. For example Darth Vader, whose face is pretty much intact but the rest of his body is in tatters and needs a sealed system to survive.
The Grand Master
The Hounding Foe need not make an actual appearance, he could just be operating behind the scenes, sending his minions. A combination can work too. You can have a Grandmaster pulling all the strings who sends a faceless evil or invincible foe against the party.
Making it Work
I think one of the most important things to making this work is avoid going too cinematic. It is tempting to try because there are so many examples from film, but I find that leads more to railroading or heavy-handed pace-setting. I've mentioned this many times before and I think it is very apt here: play fair, let your villains die when the dice say they die. Not every Hounding Foe is going to play out as you imagine it before the game. Sometimes the players circumvent them or counter their schemes. When this happens, allow it. Much of the fun from the player side is being able to test their mettle against a real threat that is trying to harm them on a regular basis. Don't take away victories just because it doesn't feel climactic enough.
Another thing that pays off here is putting a lot of thought into your Foe's strategies and tactics. Don't just have them show up thirty feet from the party every time they meet. Unless it is just a mindless killer, which in some cases it may be, plan something a bit more complex. Think like a PC here, not an NPC. This individual really wants to stop or kill the party, so from its perspective what methods other than just showing up and swinging a sword would help it achieve that. Think in terms of ambushes, coercion, trickery, and red herrings.
Also think in terms of real movement. Some Hounding Foes will have supernatural abilities that allow them to skip around, if they don't plot movements out just as you would for PCs, so it doesn't feel like they are just showing up when you decide. One method I've mentioned before is the Foe Map. This is a handy way to track the movements of key NPCs during play. I don't use it every single session, but I do when someone is after the party. Personally I find it very helpful.
Finally, it is handy to avoid overplaying Hounding Foes or using the same basic one every time. If you always use send masked Imperial Eunuchs after the party, that gets old.