In some versions, it could be through a chance encounter with old SHIELD files. Maybe he stumbles upon some classified documents while helping out at the new Avengers facility. The files might have been misfiled or left unguarded for a moment, and his inquisitive nature leads him to dig deeper.
Well, it could start with Peter needing a place to stay after some trouble at his own home. Clint, being the good - hearted Avenger he is, offers Peter a room. Then they could have a series of adventures together, like dealing with small - time criminals in their neighborhood. Maybe Peter helps Clint with his archery skills using his scientific knowledge, and Clint teaches Peter some survival skills in return.
One possible fanfiction could be found on Archive of Our Own. You can search for tags like 'Peter Parker', 'Clint Barton', and 'Living together' to narrow down the results. There are often many creative fanfictions in the Marvel fandom that explore unique living situations for characters.
One challenge could be the difference in their lifestyles. Peter is used to the city life in Queens, while Clint is more of a loner who has been on various missions around the world. Their daily routines and environments are so different that it might be hard for them to find common ground at first.
Laura Barton is a supporting character often associated with Clint Barton, also known as Hawkeye. She's depicted as a strong and supportive figure in his life.
I'm not entirely sure specifically what 'barton name fiction' refers to. It could potentially be a fictional work related to a character named Barton. Maybe it's a story created by an author with Barton as the main character or a story set in a place called Barton.