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.
Barton was a superhero from Marvel Comics, codenamed Hawkeye. He was an ordinary human without any superpowers or black technology equipment, but he was an outstanding archer and combat expert. Barton was played by Jeremy Renner in the Marvel Cinematic Universe and made his debut in Thor 1. He was a member of the Avengers and had participated in many battles, including the Battle of New York and the Battle of Sokovia. After his retirement, Patton was involved in the hero's civil war and fought on behalf of Captain America. His character was highly anticipated in the Marvel Universe, but he had relatively few scenes in movies and no independent movies. At present, Barton's future development in the Marvel Universe was still unknown.
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.