How come Bootstrap Turner (Bill Turner) ends up on the Davy Jones ship?

This is the conversation between Jack Sparrow and Bootstrap Bill when the latter came to deliver Davy Jones' message and the black spot:

Jack: And to what do I owe the pleasure of your carbuncle?
Bootstrap: He sent me. Davy Jones.
Jack: So it's you, then. He shanghaied you into service, eh?
Bootstrap: I chose it. I'm sorry for the part I played in the mutiny against you, Jack. I stood up for you. Everything went wrong after that. They strapped me to a cannon. I ended up on the bottom of the ocean, the weight of the water crushing down on me. Unable to move. Unable to die, Jack. And I thought that even the tiniest hope of escaping this fate, I would take it. I would trade anything for it.
Jack: It's funny what a man will do to forestall his final judgement.
Bootstrap: You made a deal with him too, Jack. He raised the Pearl from the depths for you. Thirteen years, you've been her captain.

So, Bootstrap was at the bottom of the ocean where, at some point, Davy Jones came and offered him the service as a way out.

This must have happened at some point in time before the end of the first movie, or Bootstrap would've just died (and floated to the land of the dead, as we saw those souls do in the first half of the third movie), which would've been his way out and he wouldn't have been desperate enough to accept Davy Jones' offer.

There is a common sailors' phrase: "Davy Jones' Locker" which means "the bottom of the ocean":

Davy Jones' Locker, also Davy Jones's Locker, is an idiom for the bottom of the sea: the state of death among drowned sailors and shipwrecks. It is used as a euphemism for drowning or shipwrecks in which the sailor(s)'s and/or ship(s)'s remains are consigned to the bottom of the sea (to be sent to Davy Jones' Locker).

The origins of the name of Davy Jones, the sailors' devil, are unclear, with a 19th-century dictionary tracing Davy Jones to a "ghost of Jonah". Other explanations of this nautical superstition have been put forth, including an incompetent sailor or a pub owner who kidnapped sailors.

So, Bootstrap Bill Turner ended up with Davy Jones because that's what happens to all drowned sailors (at least, that's what happens in the world of Pirates of the Caribbean).

From farther down the Wikipedia article:

The concept of Davy Jones was conflated with the legend of the Flying Dutchman in the Pirates of the Caribbean film series, in which Davy Jones's locker is portrayed as a sort of purgatory, with Davy Jones being a captain assigned to ferry those drowned at sea to the afterlife before he corrupted his purpose out of anger at his betrayal by his lover, the sea goddess Calypso.

In the film Davy Jones is supposed to be ferrying souls lost at sea into the Locker so they can be at peace.

However, the Davy Jones in the film is not doing his job properly, hence the weird/fish/ocean/coral like appearance of him and his crew.

Instead, Davy Jones is attacking pirates and sailors and offering them a choice, to die or to serve on his ship.

Davy Jones: Do you fear death?

This is how bootstrap Bill joined the ship. He was given the same choice.

"They strapped me to a cannon, I ended up on the bottom of the ocean, the weight of the water crushing down on me. Unable to move...unable to die, Jack. And I thought that even the tiniest hope of escaping this fate...I would take it. I would trade anything for it."

This is also why Tia (Calypso) says to Davy Jones:

You have corrupted your purpose, and so yourself.

When Will Turner becomes the new Davy Jones - he does the job properly. That is why he looks normal.

K, that might be all in well, but that doesn't explain how jones found him at the bottom of the ocean. You only can go to the locker after death... So unless he was still down there when the curse was broken, he never would have died, and there for jones never would have approached him.

So there for, he sat at the bottom of the ocean for a good ten years or so, then the curse was broken, and jones found him and made him a deal. He accepted. It might also be possible that he joined Jack's crew, shortly after Jack made a deal with jones and had raised the burnt arse lusty wench from the depths and renamed it the black pearl, only to lose it two years later to Barbossa.