Miss Fortune's family was murdered when she was a child. She grew up on the streets of Bilgewater, and became a pirate hunter. People sometimes call her just a mercenary or a bounty hunter, but no. She is very specifically a pirate hunter. She wants to rid Bilgewater of piracy, and fights to clean up the city and make it a better place.
Gangplank is a pirate whole father MAYBE was the one who killed MF's family. There's not too much to say about him, really. He is nicknamed the Saltwater Scourge, and he is the most feared pirate in Runeterra.
Graves' early life isn't really delved into, as far as I know. He was born in Bilgewater, but most of his lore focuses on what happened between him and Twisted Fate.
I'm pretty sure Nautilus knows his family is dead and likely so is his former crew, but yes, his mission is still to find them all.