Homeschooling allows for something that simply can not be gained in a public school setting, and that is the opportunity for the parent to point toward examples of good character on a daily basis...heck, on an hourly basis! There is no "Good Character" campaign with slogans and posters taped up in school hallways that will ever be a good substitute for the conversations and intentional parenting time that homeschooling offers.
Now, don't get me wrong, I am not saying that parents who don't homeschool can not instill great character in their kids, but it is far easier when you have opportunities all day long.
Building character in our kids happens in a multitude of ways. It is brought in via the reading selections used for literature, it can be tapped into during writing assignments with moral dilemmas presented to ponder and write about, and it can be discussed when talking about world events and the actions of national leaders and famous folk are on display. Character can be explored around the kitchen table as the actions of family and friends are used as examples of (hopefully) good character.
Some parents bring in character curricula, or take an individual positive character quality to discuss each day. Many use faith based tools to teach the various qualities found in good character that they hope to instill in their children.
But you know what the single best character development tool is?
How you and your spouse walk through the world is watched and noted. It is copied from the youngest of ages, and we all had that surprise awareness when you saw our littlest ones imitate something we said or did. Well, that doesn't change because your children are older, it is just that they are more self-conscious about it and don't eagerly parrot you any more. No, instead they take it all in, they begin to evaluate your examples, and they make decisions about who they want to be in the world based largely on who YOU are in the world.
How do you handle frustration? Are you trustworthy and honest, or do you often tell little white lies that grow over time? Do you set an example of humility? If your kids walk in your footsteps, will they be viewed as having a strong work ethic? Do you reflect qualities you'd like to see develop in your children? Are you kind to everyone? Do you treat waiters with the same respect as you treat business leaders?
There is no novel, no film, and no discussion that will have as large an impact on what kind of adult your children mature into than watching your own silent example. But supporting that role modeling with conversations, and the intentional selection of good moral character exemplified in what they read and watch can continue to speak to them where your role modeling leaves off. But always remember, anything that is curriculum oriented is secondary to your own example.