Hodrick says the Mound-Makers are a group of madmen and hollows that love to pile up their victims and collect their vertebrae. Whether there is any purpose beyond that is unknown; they don't seem to be affiliated with any god or have some endgame plan for their collection, so for now its safe to assume they just love to collect vertebrae. They will try to kill everything around them, and they are also in rebellion against the gods in a way. All of the other covenants have some kind of god or god-like being, you essentially are giving up your agency as a being to whatever god you choose, and can only strictly act in a way of that god's choosing. One of the Mound-Makers victory conditions is killing a phantom who are servants to the gods, and killing them rewards a vertebra shackle. The Mound-Makers serve no gods however, and can act in any way they choose.
Holy Knight Hodrick is a former knight of the Sunless Realm, the grandfather of Sirris, and a certified Mound-Maker and madman. He is crazy. His granddaughter, Sirris, is on a quest to kill him, presumably because of his madness. Sirris is a self-proclaimed former knight of the Divinity, possibly a Darkmoon knight of Gwyndolin, and is or was a knight of the Sunless Realm and a servant to the Nameless Moon. When she says "former servant" of the divinity, she may also be speaking of the nameless moon considering she left to kill her grandfather. You could probably speculate either way that she was a darkmoon knight, or is/was a sunless knight.
So Hodrick, a former knight of some kind of divinity, is now a free agent. A violent, crazy agent, but still free from the whim of the gods. His granddaughter, who is or was a servant to some kind of divinity, has come to kill him, but more-so for personal reasons than because of political reasons like being sent by divinity to quell the crazy rebel. Maybe there is some poetic justice in the vertebrae description where it states that 'each victim is another connection, an addition to the family' and then Hodrick is killed by a literal family member.
The Mound Makers aren't necessarily in open rebellion of the gods, there isn't some political movement to overthrow them and set up a republic or anything. They are a gathering of unique individuals who have gone insane because of their circumstances. Undead or hollows who have embraced the freedom that insanity brings and now just lash out at the world the gods have created, 'freeing' one individual at a time and creating a larger family.