We do not know enough about Han and his behavior PRIOR to his first appearance to reliably assign him an alignment. At best we can ASSuME some things based on what he says regarding his philosphies to Ben, Luke, and Leia in Episode IV, but none of the movies EVER delve into Han's past beyond his involvement with Jabba which says almost nothing about him. As Umbran noted he TALKS very neutralish, but TALK is not what determines alignment. Actions determine alignment.
When introduced to Han we learn that he is a smuggler. He's someone who makes his living operating outside the law inside of a fascist dictatorship (the Empire) just reaching the height of its power. Then, when Han shoots Greedo (whether shooting first or not) it IS self-defense. Greedo has a gun pointed at him and in an unambiguous conversation Greedo makes it clear that HE intends to kill Han - and has been looking forward to it - even if it is not what JABBA wanted Greedo to do. But although it's a dramatic choice of action to shoot Greedo first on the sly under the table it CANNOT show a trend yet.
The next real moral/ethical choice that Han gets to make is when Luke has to promise him money to get him to help rescue Leia. Now this seems quite selfish (and neutral) but we ARE talking about a guy who, given the circles he's running around in, HAS to put up a very neutral, selfish front lest he be thought weak and unreliable. And remember that for a scoundrel and smuggler, and even for all his bragging - HE'S NOT VERY GOOD AT IT. Obi Wan's reactions to him in the cantina tend to support this, as if he's sensing how Good-aligned he really is under his blustering facade.
Han owes Jabba a ton of money. Though it is never stated just how much he owes, it can be inferred that the 17,000 might "really save his neck" but it won't necessarily fully square with Jabba what he owes. In the conversation with Greedo the lost shipment that is mentioned could be where the total of the debt comes from, but given the overall context it seems FAR more likely that this is just the latest in a string of incidents that have left him further and further in debt to Jabba.
So, when Han hears about the Princess his reluctance to get involved can still be seen as ranging from neutral to NG or CG - still not enough evidence of a pattern of actions to base a reliable alignment estimation upon. But he NEEDS that money, and if it's not the only motivation he has for helping at that point it certainly helps him give in to his nicer tendencies. Also, Obi Wan gave him 2000 up front and promised 15,000. That shows either that Han is just that desperate or that he's NOT the hard-assed Neutral he pretends to be and is thus willing to take Obi Wan at his word. Now aboard the Death Star, Luke promises "more wealth than you can imagine". Han HAS to know, given that he's currently trapped on this fascist-dictatorship-operated space station large enough to be mistaken for a FREAKIN MOON, that living to see any of this fantasy money is NOT a real reason to get motivated to rescue princesses. He uses the money as a convenient excuse to maintain his persona and yet do what he senses is the right thing - fight the Empire, save the girl.
The real crisis of conscience for Han is at the rebel base when he gets his reward. Although he's now FREE of the underworld life that he'd gotten himself into he's likely been too long in the habit of acting Neutral or CN (and thus effectively being Neutral/CN) to make the alignment shift yet. But it's not long before he changes his mind - and possibly his alignment - and comes back to help save the day.
NOW, at THIS point, you can start to throw alignments on him. He's not neutral - even if he once was. He's clearly shifted towards good (or established that he's always been so, even if he's had to pretend otherwise), though his old hard-dying Neutral behaviors for appearances sake keep him NG or CG. In Empire Strikes Back he's VOLUNTARILY going to go back to pay off Jabba the Hut. He says because if not he'll be a dead man, but he's not a criminal anymore. Why should he feel obligated to pay a debt to a criminal who might indeed want to have him killed rather than have the money at this point? He wants to pay Jabba because Han has a sense of honor and honesty that he cannot continue to deny. An obligation willingly accepted must be fulfilled. No matter what his motivations were at the time he incurred the debt his alignment NOW will not allow him to try to welch on it - and he could just as easily borrow an x-wing and a few troops and put an end to it and the Rebels might well GIVE them to him if he asked. By the time of Return of the Jedi those neutral tendencies are all but entirely eliminated and he's easily seen as LG - he's not even taking it for granted that his faithful companion Chewbacca will just go along with whatever dangers he's getting himself into.
YMMV