You are right that Louis XIII outlawed dueling.

However, you have to distinguished two different types of duels. The one that Louis XIII outlawed was what was referred as duels judiciaires or judicial duels.

This kind of duel was extremely old ( well before Charlemagne ) but a common practice during the Middle-Ages. The two opponents and a third party ( being a referee ) agreed to the rules such as duel to death, first blood or whatever. The loosing opponent, if alive, suffered a punishment according to the law. Only the king could grant pardon. This practice was also considered to be a divine judgement. For example, if a crime was impossible to solve and that the person accused of X thing ( rape, murder or whatever ) was impossible to be cleared or proven guilty such duel could be organized. The outcome would be considered as the judgement of God and again, whoever lost would then be sentenced to the punishment that fit his crime, unless granted a pardon.

If Louis XIII definitely put an end to this tradition, no judicial duel was fought since 1547 ( at least not one that has been authorized by the monarch so the outcome would not have had judicial consequences ).

As decades went on and the royal power gained more and more power at the expense of the nobility ( At that time France was heading toward absolute monarchy ) new ways had to be found to " fight " this royal power. The transition to a new " dueling system " was partly due to this eternal struggle between the monarch and the aristocracy. Since dueling in order to have justice was forbidden, the aristocracy had to find a way to break this new rule and that way was honour.

The duel of/for honour was actually quite a mess ( beg your pardon for the term, but it fits ). Aristocrats invoked any reason to fight each other in duel. Thinking someone harmed your honour or your family's was enough to start a duel. The reasons could be quite serious of course but sometimes they were incredibly ridiculous ( you looked at me in an arrogant way then you shamed me, ...Etc ). And of course these duels were also pretexts simply to fight someone you did not like, to prove you were better at swordplay,...Etc

At the end of the XVIth Century and beginning of the XVIIth, dueling for honour became a trend. The youth played a huge part in spreading this and it became a serious issue for the government. You could almost compare it with today's drinking/driving issue. You had this youth fighting each other, getting killed or harmed for no reason. And the numbers were huge. My brother-in-law who does fencing confirmed me that according to Pierre Lacaze's From duel to fencing some 10,000 men from good families died during the two decades before 1608. It simply had to stop.

Between Francois I and Louis XIII, the kings passed several laws to prevent this but they were barely enforced until Cardinal Richelieu and Louis XIII passed an edict forbidding duels on pain of death. Dueling was now a crime of lèse-majesté ( disobeying a royal order/law ). Many members of the aristocracy protested against it and an arm-wrestling between them and Richelieu started. Richelieu had the final word ( legally at least ) :Kings are made to protect the lives of their subjects. One or two people were executed as an example but most people were simply locked for a while or sentenced to pay a big fine. Louis XIV indeed thought he could make it all stop but it was not the case. At best, duelers refrained themselves from killing each other but people kept dying all the same, though at a lower rate. The later XVIIth and early XVIIIth century brought dueling to a new level actually :

It became a symbol or aristocratic and bourgeois defiance of the absolute monarchy. As I said earlier for many duels were : " *You said we can't do it but we'll do it anyway ". It also became a romantic symbol, clerics entered the game but more astonishingly women as well ( such as Julie d'Aubigny whom I think was the subject of a question here last week ). Louis XVI also tried to stop it, passing more laws but less severe ( people risked being exiled ) and the end of the monarchy signed a renewal of the practice in France ( the last duel was in 1967, I talked about it here ).

I hope this helps you a bit. You asked Would the families of the deceased press for charges or were they more likely to just perceive the death-in-duel as acceptable. The answer is no, not that I can think of simply because the deceased were themselves out of the law by dueling, even if they died. However, this is how blood feud started as personal revenge was a way to " avenge " someone killed. You can imagine that this created even more legal problems for everyone.

Sources : The Great duels that made France by Alexis Brezet and Jean-Christophe Buisson

Duels, a French passion by Jean-Noël Jeanneney.

I unfortunately can't think of a book in English specific to the subject. If you happen to read French, in 1895 Marcel Andriveau wrote a thesis about dueling in France ( during the Middle-Ages though ).