I hate the fact that people keep saying that Han shot first, when you go frame by frame, you can see that Greedo shoots first--misses entirely!!--and then Han shoots with ease right at Greedo's face.
Han is a great smuggler, but he never shot first. That's like saying Chuck Norris could force choke a storm trooper from seven planets and not lift a finger (made up on the spot).
You're talking about the newer version though. In the original theatrical release, Han doesn't do a weird head bob thing and Greedo is simply threatening Han. Why go to all the trouble of showing close up shots of Han unholstering his weapon under the table if he's not going to try to surprise Greedo.
Also, how could Greedo POSSIBLY miss a stationary Han at less than three feet away? He would have to be completely inept.
I think that Han, at the beginning of the story, was a experience hardened criminal looking out for his own skin. He smuggles things for a living, he's going to shoot first to get away with his life. He's a reluctant hero who has suppressed his interest in doing the right thing in favor of doing the profitable thing. But the adventure pulls his goodness back to the surface. For a while doing the right thing was simply in line with saving his own hide and making a quick buck, but later you see him come back all on his own and do the right things in spite of his smuggler tendencies. It hints that he is recovering the man he once was.