D&D 5E - 5e Vampire Hunter/Slayer - How would you do it?

Personally, a straight-up Vengeance Paladin would seem to make most sense to me. Possibly with the addition of a (custom) feat to make it just a tad more Hunter-y. The dex focus is easy (there being 0 barrier to dex focus for most melee classes). The Vengeance Channel Divinity options seem great--Vow of Enmity is an accuracy-based alternative to Hunter's Mark, and thus better synergizes with the Paladin's answer to getting extra attacks (that is, getting auto-damage on every hit), and Abjure Enemy seems like a superior alternative to Turn Undead when dealing with individual powerful undead enemies, rather than a horde of them (which is rarely how vampires roll--and when they do, the Conservation of Ninjutsu generally means they're much less dangerous foes.) The Oath spells include Hunter's Mark, which means you always have it prepared in addition to whatever other Paladin spells you want--and eventually you won't really need those first-level slots for Smiting anyway. Plus, the Paladin's extra damage features hit "fiends and undead," which includes vampires, for more damage anyway.

Then, if that doesn't seem enough, you could try for a custom feat--Vampire Slayer, of course. It could grant Turn Undead (with some kind of special bonus against Vampires specifically) as an additional CD option, and also the Ranger's Favored Enemy feature, but exclusively for Vampires. Oh, and of course it could always include a special "Chain Whip" proficiency, that is identical to a regular whip but better damage dice (probably 1d8, but at least 1d6), for that truly Castlevania flavor. I dunno what else one could do for the feat, perhaps some kind of special ritual (or other method) that destroys a vampire's ability to revive after being slain? Individually, none of those benefits really feels worth a feat, but all of them might be a bit strong for a feat--perhaps pick two or three. Just suggestions, either way.

From there, the only other consideration I could possibly think of would be to try to persuade your DM to let you swap Charisma for Wisdom for all your Paladin class features. Then, you'd be a high-Dex, high-Wis holy warrior with a specialty in hunting and destroying vampires.