In the comics, Black Widow and Hawkeye were most definitely an "item." The two are introduced as lovers in the same comic series, five issues apart, Black Widow was still working as a Soviet spy at the time, and Hawkeye was introduced as a villain.
On the run, the naive Hawkeye met the Black Widow, a spy for the Soviet Union, with whom he fell in love. Blindly following the Black Widow, Hawkeye aided her attempts to steal technology developed by Tony Stark.
Later, Black Widow goes on to have relationships with several other male superheroes, notably Tony Stark, but her and Hawkeye have a continuing 'thing' between them. Hawkeye also pursues other relationships, most notably with Mockingbird, whom he marries.
As far as the movies go, it is somewhat obvious that they have a connection, but it has been kept deliberately. This is possibly because Marvel is saving their relationship for a future Avengers or stand-alone film, or possibly because they have decided on a different direction.
While Captain America does have more than one love interest in the comics, Sharon Carter is always the main one, and I do not know of Black Widow and Captain America ever pursuing a relationship at any point.