In the comics, he became the new government-sponsored Captain America after Steve Rogers quit, but snapped after his parents were killed by terrorists and murdered them in cold blood, then relinquished the shield back to Steve. After a brief recovery, he became the U.S. Agent, still loyal to the government.
He's generally portrayed as an abrasive but ultimately noble man who, despite somewhat skewered political views, wants to do good and recognizes he is not good enough to be Cap. The show seems to be going the same route, emphasizing that he is a family man devoted to his wife and daughter and a dutiful friends to his old army buddies, even recruiting one who is going through difficulties to become his sidekick.