TL;DR: The Grinch is the sole survivor of a purge by the Whos, wiping out all nonbelievers. At the end, he's forced to convert in order to escape his life of poverty and seclusion.
First off, there's his appearance. He's clearly physically different from every other Who: taller, green fur, yellow eyes, etc. He's a member of a completely different race than the Whos... but there's no other member of that race in sight. We also know that he's not just some genetic aberration -- the cartoon version mentions he gains the strength of "ten Grinches plus two". Not ten times his own strength, but the strength of ten people like him (plus two).
So, where are the rest of his people? Simple: he's the Grinch, not a Grinch. They've been wiped out. Every last man, woman, and child, besides this one lonely survivor.
But who (pun intended) could possibly have done such a horrible thing? Who (pun also intended) do we know that have a near fanatical belief in their religion? Who (this pun was completely accidental) makes up 99.9% of the population, and controls all the resources? The Whos.
For some reason, their religion caused them to wipe out the entire race of Grinches, leaving only one alive. It wouldn't be difficult -- the Whos have massive numbers on their side (one man had 96 daughers), and for a supposedly peace loving people, seem to have some giant knives (as well as giving their kids javelins and bows). It also would be believable -- one of the few things we know about the Whos is how super religious they are. Them deciding to go all "Deus Vult" on the Grinches for refusing to follow their religion wouldn't be hard to believe.
The Grinch survived (likely due to him being frozen in ice, only to be thawed out 100 years later by two siblings). More realistically, he likely fled, or was already living as a hermit, allowing him to escape the slaughter. He lived at the top of a remote mountain, and due to his cunning, likely had a number of traps and defenses set up. The Whos were content to ignore him, so long as he was out of the way.
So of course the Grinch hates the Whos for killing everyone he ever knew (it's mentioned that he hates them all year round, not just at Christmas). But Christmas makes him even more enraged. He wants to forget about them, but on that day, the noise they make causes it to be impossible for him to do so. Even more enraging, they're singing religious hymns of the religion that killed his people. It's highly possible that Christmas was the anniversary of the Grinch Genocide.
At the end of the movie, the Grinch is shown making a change for the "better". Did he though? Notably, both the book and cartoon state that "some say his heart grew three sizes that day". It doesn't just describe it happening, it states that certain people would make that claim. It's Who propaganda, to cover up the harsh truth: The Grinch had grown depressed, and knew the Whos would kill him for ruining their special day. He was open to death, especially if it involved screwing them over, and letting them know the Grinches would never give up. However, at the last moment, he had a change of heart -- not out of love, but self preservation. He decided to assimilate instead, and pretended to join their religion. That allowed him to not live in a cold cave, and also granted him a measure of security. If he pretended to be a convert who was truly inspired by them, and declared so publicly, the Who religious officials couldn't kill him without public backlash.