I've been inspired by the new League of Legends champion Naafiri, where her gameplay fantasy can be described as "alpha of a pack of hellhounds." I was looking through the subclasses I own in the DnDBeyond database, but couldn't find one that really fit the bill. Does anyone know of any homebrew subclasses that might let you fulfill that kind of roleplay fantasy in D&D?

It's not homebrew, but you could reflavor a Shepherd druid with Conjure Animals to get 8 wolves or 2 dire wolves (and more with upcasting).

Actually using the hell hound stat block would be pretty tough to balance. They are CR 3, and for reference a Moon Druid can't shift into a CR3 creature until level 9. Summoning 3 hellhounds would roughly be equivalent to an 8th level spell, so you're either doing something else for a whole lot of levels or you're reflavoring a weaker option like regular wolves.

I could see a subclass with an evolving statblock like the drakenward’s drake. It would start out as a “reduced threat” hell hound and then evolve into a fully fledged hell hound and eventually into a more powerful hell hound. I could even see the subclass summoning a small pack of standard hell hounds for a minute at higher levels with the companion as the alpha or something. That could be cool. The only real problem with it is that hell hounds only exclusively serve evil beings.