If there were a game where your character had to protect family / children then I think alignment would not even matter.
Actually I think this example could be used to sort of focus the discussion on alignment in a way. Different alignments would perhaps place different values on protecting their family and just how far they would go to do so. In my opinion alignment may be an overall outlook but it shouldn't be used to limit a players options/reactions. Rather it provides a backdrop perhaps for some internal conflict whenever obstacles or threats are presented.
While the good and evil elements of the alignment spectrum seem to speak to what the person values, whether their value is centered more on self or on others and whether they value the lives of others or not; the lawful/chaotic element of the alignment spectrum deals more with the means and/or methods they use to achieve their desired end and what they may be willing to do to achieve the thing that they value.
Most characters would want to protect their families but I think their alignment may sway them one way or the other in terms of how they will protect their family, and how far they are willing to go. Alignment might also provide some insight into what value they may place on their family and why. A lawful evil character may want to protect his/her family because the evil element of the alignment tells him that the family is an extension of himself or that they perhaps "Belong" To him and so a threat to his family is threat to himself. Meanwhile the lawful bent of his alignment suggests a character who either places some value in some form of creed or tradition (Perhaps the idea that family is everything), but it could also suggest the methods that this character will use to protect his family are going to utilize the law or cultural norms of whatever culture he finds himself in. He will perhaps manipulate the system so that it works for him and his family, and against the person or entity that threatens it. He is perhaps more than willing to kill but is more likely to do so using the system, perhaps framing his opponent for murder and having them executed as opposed to killing them directly.
Someone of a chaotic bent would perhaps be more likely to react in a more direct and violent way against a perceived threat against their family; perhaps directly killing the person threatening them. They would be less inclined to act within the law or some form of creed of conduct in order to protect their family. They might kill, steal, assault, or otherwise bring ruin upon that which threatens their family. While this doesn't mean that a chaotic character is stupid or incapable of planning ahead or that they will act upon any inclination regardless of consequences, they are much more likely to do something that violates societal norms if they believe that they could get away with it or if they believe that the consequences for doing so are minimal. They may not murder the person that is threatening their family out in the street, but if they get that person into a position where they can kill them without being caught, they are more likely to do so.
I think those with a chaotic bent are more likely to have a "Might makes right" Philosophy and use force or coercion as a means to achieve their goals, but they are not limited to these methods. While the lawful evil character may use/manipulate the system and the culture to protect his family, the chaotic evil character is more likely to take more direct means and use force to protect their family.
Interestingly enough the chaotic good character may react in a manner fairly similar to the chaotic evil character in that they may be more likely to use force than the lawful good character or even the lawful evil character to protect their family, the value that they place on their family and on life in general is different. The evil character does not value the lives of others, in general, and if he does it is only in relation to himself, either as extension of himself in some way or as a possession. The good character does place value in life and if he can protect his family without causing pain, suffering and death in others he will do so.
The lawful good character will likely operate within the law to protect his family and if he can stop the threat without killing the person that is causing the threat he will not kill them, nor will he be likely to kill someone who may have been threat to his family at one time, but is no longer a viable threat unless that person has been sentenced to death by some form of legal authority. He will want to find a legitimate way to protect his family legally while the chaotic good character may not concern himself with whether the means by which he removes the threat is legal or socially acceptable.
Of course discussing alignment is not unlike discussing a sociological theory in that it is often uniquely applied to each situation without much in the way of hard and fast rules. I have also seen chaotic/lawful discussed in terms of freedom vs. Restraint and the concepts of good/evil in terms of love of others vs. Love of self at the expense of others. It's a pretty deep rabbit hole really.