Technically speaking the arc to Han's character begins and ends in the first film.
He was a mercenary but through interaction with a young idealist he becomes part of something larger.
ESB winds that back a bit by having Han constantly reminding us that this is a temporary measure and he has outside troubles that need attending to. He serves more as an instrument of Leia's character arc as she goes from being a dedicated leader of a resistance army in the first film to the sort of person who just abandons her cause to rescue her boyfriend in ROTJ (the Skywalker thing may have been tacked on but at least it plays out using the family MO).
She is attracted to Han because he is unpredictable and irreverent (and rather dishy). There could have been a lot of mileage left in what happens to Leia and Han if they settle down, if he stops being a scoundrel ; if he stops being so dishy (imagine if the carbonite had permanently scarred him) ; if he was vulnerable (no longer able to be the flyboy pirate because he is permanently blind and bitter because of it).