Your eyes sparkle when you're angry
One of the most exaggerated characteristics of Clint Barton is his role as a ladies' man. In fact while Clint's confidence is initially effective for wooing women he does not have the ability to follow through and often ends up disappointing any women he becomes intimate with.
The first woman that Clint becomes romantically involved in during his time as a hero is Natasha Romanoff, or Black Widow. At the time Natasha Romanoff was working for the Soviet Union as a spy. Her relationships were all the effects of her training rather than a real romantic connection. When she was granted amnesty for her crimes he went a step further and covered up her past ruthlessness in an effort to get her into the Avengers (who had a specific code of ethics regarding killing). She refused the offer and broke Clint's heart. From there Natasha and Clint were still a strong team while working together but they were never romantically involved after that point. That being said Clint calls her his work wife.
Arguably the important woman in Clint Barton's life was Barbara 'Bobbi' Morse, or Mockingbird. She started as a S.H.I.E.L.D agent and biochemist who initially was recruited as part of an effort to recreate the super solider serum. While at S.H.I.E.L.D. she began going around as a hero by the name of Mockingbird. Early in her career she encountered Hawkeye and teamed up with him to bring Captain America out of a short lived retirement. During their second encounter they were kidnapped by Crossfire who was testing hypnotic ultrasounds as a way to force the Avengers to kill one another. The two heroes were forced to beat each other until Hawkeye removed the arrowhead of his sonic arrow and triggered it while it was in his mouth causing significant damage to his hearing. The experience the two heroes shared led them to elope. Bobbi joined the West Coast Avengers team, led by Clint at the time and for a while they were the team's guiding force. The stress of their work and Clint's inability to reconcile Bobbi's refusal to stop a villains' death that she could have prevented with the code of conduct of the Avengers led them to separate. The two still work together regularly and actually make quite a team but all attempts to rekindle their relationship have failed.
Hawkeye and Jessica Drew, aka Spider-Woman grew close after an incident where they battled Nul in Brazil. Following that event they went on several dates that often turned into missions where they fought against the likes of Hydra, AIM, and Norman Osborn's H.A.M.M.E.R. The romance was doomed to fail however because Clint's self-destructive tendencies and inability to consider the consequences of his actions. Clint cheats on her in a one night stand with a girl he saved which leads to their break up.
Clint also showed interest in Scarlet Witch at one point but she didn't return his feelings so nothing ever came of it at that time. He did seek her out when he returned to life after she had killed him when her powers grew out of control. He found her in a remote village in Europe with no memory of her life and during that time they did partake in a one night stand.