After you have played D&D for awhile you start to hate chaotic neutral, especially if you are in the Dungeon Masters Chair. People pick this alignment to have no restrictions and still not be evil. The effect is that the character is assumed to have no alignment, which sucks for folks really trying to play within the confines of another alignment. Don't get me wrong dealing with lawful good can suck just as bad, though it usually is the other players complaining. As a person who naturally falls toward the chaotic side of the spectrum I want to make the alignment become more nuanced.
The alignment descriptions of the Chaotic always include a desired for freedom and usually mention a rugged individualism. That is nice but not much to go off of. Chaotics' are people that have a general suspicion about groups. The saying, "the needs of the many outweigh the needs of the few" is a lawful statement. Chaotic see groups as devolving to their lowest members or harmfully inflicting their will upon people. They will see themselves as apart from society and will often be apart. Think alones not wanting or accepting help from others. They may even feel help from others comes with strings attached. They are possibly introverts or loners walking a path outside the norm, perhaps consciously so. They are not going to need acclaim or acknowledgement of their works, they do it for their own.
Chaotics' are going to be disorganized people. They will go to the store without a list and wing it. They are going to try things without the proper tools or and will trust in their own skills. When dealing with too many instructions they may throw up their hands and proceed without them. You will chafe at restriction on what you can do, even if not actively fighting with the rules you will have a hard time living under them. While chaotics' may not all be introverts their view of the world radiates out from the self. Perhaps family trumps the individual, but their commitments go from small to large. As a chaotic person I want my daughter to succeed even at your children's failure. I want my friends to get jobs before others. I want my community to prosper and finally my nation. The bigger picture comes last. We will be people with fewer but deeper interactions with others. Maybe fleeing from large gathering in favor of being one on one with people.
I read a post saying "Chaotic people think they are above the law", nothing could be further from the truth. Chaotics' disagree with laws, thinking them unfair, stupid or unjust. Chaotics' will not follow laws for their own sake. If the law can not be enforced and gets in the way of a chaotic person it will not be followed. The chaotic person will understand potential consequences can be applied. As a bit on politics. Someone on a post thought Ron Paul was lawful because he always talks about conforming to the constitution, this is absolutely incorrect. Ron Paul is a Chaotic Poster Boy, as were the founding fathers. Any government that states individuals have rights that exist before the government and can not be taken away by that government is chaotic in nature. Most folks want to see their political party as good and the opposition as evil, but lets be real. Socialism, Communalism and Fascism are Lawful forms of governance. Anarchy, Antifederalism and Libertarianism are Chaotic forms. Philosophies like Objectivism and Existentialism are Chaotic in nature.
Your chaotic characters are going to have weird rules that guide them. Thinking of The Hound in A game of thrones, he is willing to hurt and kill people over chicken, but will keep young girls safe. Sandor seems to have protecting girls and will never work for fire users as his guiding rules. He threw away a desirable position with the Lannisters when Tyrion used fire at the blackwater. Chaotics may also make choices that are not in their best interest. They will follow a path that feels right to them even when better paths become available as other paths may include compromises they are not willing to make or alterations that do not feel comfortable for the chaotic person to live with.