There are two heroes currently using the name Hawkeye in Mainstream Continuity.
Clinton Barton was a former circus worker and criminal-turned-hero, who received his archery skills as a young boy through training with Swordsman and Trickshot. Barton would eventually join both the East Coast and West Coast branches of The Avengers, even leading at one time. Hawkeye was created by Stan Lee and Don Heck, first appearing in Tales of Suspense #57 (1964).
Katherine Bishop was a regular civilian, with considerable archery skills, who happened to cross paths with the Young Avengers on a mission, eventually inviting herself onto the team. Despite Barton's return as Hawkeye, Bishop still operates with the name and even became his protégé. Katherine Bishop was created by Allan Heinberg and Jim Cheung, first appearing in Young Avengers #1 (2005).