In this chapter, a Levite joins the household of Micah to minister to his idol.
This is a pretty crazy chapter, and the first chapter that contains the expression, "there was no king in Israel, every man did what was right in his own eyes." This is an important verse to Judges, which both foreshadows the future kingdom and also tries to explain why the kingdom is necessary. This phrase, and even this entire book, point out the sinful tendencies of the Israelites in a bunch of ways.
Already we have seen Samson sin in several ways, like eating unclean food and sexual immorality. Now in this chapter we have Micah stealing money from his parents, but then returning it out of fear of a curse. His mother celebrates the occasion by asking Micah to construct an idol, to which he complies. Lastly, Micah consecrates one of his sons as a priest, which again contradicts the Law, because the Law demands that every priest must be a son of Aaron.
When a wandering Levite passed by, Micah realizes that a Levite would be an upgrade over his son, so he hires the Levite to serve as his priest instead. The Levite is perfectly happy to offer sacrifices at Micah's idol, so he agrees. This indicts the Levite, because every sacrifice is supposed to be made before the bronze altar in front of the tent of meeting.
Nearly everything in this chapter is a sin, from beginning to end. This is what it means by "right in his own eyes"; that people are behaving as they see fit, rather than following the covenant or the principles of the LORD. It implies a sort of wild godlessness, akin to the notion of moral relativism.
The reason why this relates to the kingdom is that the king is a unifying figure in Israelite society. When Israel gains a king, he will lead the people and direct them to all behave in the same way and with the same standards. In theory, the king is supposed to espouse the law and direct the people to obey the covenant, bringing unity to the nation and directing the people so that they do what is good, not "what is good in their own eyes".
In practice, we will see that a few of the kings are good and many of the kings are bad. The good kings direct the people to follow the law, and the bad kings direct them into idolatry. Either way, much of Israelite history is dictated by the moral standards of whoever was king at the time. So it's definitely true that the kings ended the lawless period of their history, but it's not true that the kings ended their sinful period.
I once heard someone teach out of this passage, and more than anything else what he criticized was the notion that Micah could buy himself a priest and that as a result, "the LORD will prosper me". Micah seems to have the notion, popular with many, that by purchasing or constructing various trappings of religiosity, that God would bless him. He built his own idol, made his own ephod, and then hired his own priest to minister to him, and he thinks that's enough to draw the LORD's blessing. This is a very individualistic notion of faith and it runs contrary to the strongly communal covenantal structure. It's also a very materialistic notion, and it assumes that God cares more about the appearance of devotion than the heart reality of it.
As it relates to individualism, the covenant was centralized by design, with three national feasts every year where all the males were required to appear before the LORD. Sacrifices had to be made before the LORD, and the LORD was only found in one place, the tent of meeting (a.k.a. the Tabernacle). Trying to maintain his own idol and priestcraft runs contrary to the principles of the covenant.
But this is not just Micah having a personal faith. In fact, in some ways Micah's faith is very impersonal because he expects that money is going to buy him blessings, when what the LORD wants from him is devotion. That is the materialism side. Micah is cutting his own way forward, doing what is right in his own eyes. But there is a big difference between ignoring human standards in your pursuit of God and ignoring God's own standards in your pursuit of him. In my opinion, human standards are flexible. The social or cultural expectations for how people are supposed to pray or worship have changed many times and in many ways over the years, and that's generally okay. In my opinion, a certain amount of drifting away from those standards is okay.
This is what it means to have a personal faith, to pursue God in a way that is your own, praying in your own way or worshiping in your own way, but always within the broader concepts of how God desires to be worshiped. To disregard God's standards, how he tries to structure human interactions with himself, is not any kind of faith at all. If you know how God wishes for you to approach him, and you disregard that way, how can it even be considered approaching God? God does not bargain over blessings, he does not sell them in exchange for worship, because that cannot be real worship if it is done for a price. And it just doesn't make sense to construct an idol and pretend that God is okay with that when the Law clearly says that he is not.
It's like trying to buy a friend. Money cannot ever buy a real friendship. In the same way, offerings cannot buy God's blessing. God's blessing comes upon the hearts that seek him.