I always thought he got caught in some strange offshoot of the black mist (hence him not dying after sinking to the ocean floor) but he doesn't really connect to any of the shadow isles champions (he is not a true undead)
Really, some champ designers were like, what about a guy in a diving suit with an anchor, and they made him, and then remembered that he needs lore, so they threw something vague together and called it done.
I like the explore the ocean depths, find strange (probably magic) goo, get consumed and abandoned aspects. He doesn't seem too driven by revenge (only a few vague lines: "find the guilty", "they will pay", "left to die"). He seems more akin to Kassadin, a being who saw something beyond comprehension and is trying to share what he saw as best he can ("beware the depths", "do you hear them calling?", "All will drown", "Fear the ocean's weight", "Peer into darkness", "echoes from the deep")
So really, the questions are twofold
First off, how did Nautalis get to be the titian of the depths? Was he a human in a diving suit that encountered something strange beneath the waves as his old lore suggests, or is he some Herald of Nagakabouros, created or reclaimed and given new purpose?
And secondly, what drives him? What is his purpose/goal? Why is he wandering Runeterra/Bilgewater rather than wandering the deeps?