This question was asked on a private education page where I post regularly, I took the time to develop an answer.
The systematic training and development of children in the nurture and admonition of the Lord. Eph 6:4
It is the outworking of Deuteronomy 6.
Specifically these verses “Hear, O Israel! The Lord is our God, the Lord is one! You shall love the Lord your God with all your heart and with all your soul and with all your might. These words, which I am commanding you today, shall be on your heart. You shall teach them diligently to your sons and shall talk of them when you sit in your house and when you walk by the way and when you lie down and when you rise up. Deuteronomy 6:4‭-‬7 NASB

It is good to read the entire passage to see the reasoning behind why we would do the teaching, guiding and nurturing we do on a daily basis with children.

Keep in mind, if children are young and have not been saved yet, we are not separating character training from the gospel but reminding them that these are things God will develop in them if they believe in Jesus as their Savior. We discuss the basics of the gospel- I Cor 15:3-4, our sin, our need of a Savior, that Jesus, as God, lived a perfect life and died for us and rose again to show He had power over sin and death, and how we can respond in trust. We want to see them walk in obedience to us as parents and be kind to others on the outside, but more than that we want them to put their hope and trust in Jesus Christ.
Once they have trusted Jesus we can continue to encourage the development of the things listed below.

Biblical Character Education involves teaching them many things. For a good resource, I recommend Kim Sorgius has developed kids bible studies that aid this learning that you can access here: about the fruit of the spirit and how they can demonstrate those in their lives from Galatians 5:22-25
But the fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith,
Meekness, temperance: against such there is no law.
And they that are Christ’s have crucified the flesh with the affections and lusts. If we live in the Spirit, let us also walk in the Spirit.

In order to walk in the spirit that means it also involves teaching them about spiritual gifts and how they can exercise their spiritual gifts in the body of Christ.
See Roms 12, 1 Cor 12,

But more than just teaching it also involves a systematic daily intentional focus on developing these characteristics of godliness in their lives.

Suggestion: Have a character trait of the week that you incorporate into everything that you teach.
For instance if “gentleness.” Read a verse every day that had to do with gentleness.
Talk about how that could be demonstrated in daily life and then as things came up throughout the week practice the exercise of gentleness or talk about how to exercise gentleness.
For example, if another child was struggling with something rather than making fun of them or mocking them, we would come alongside and ask how we could help and be kind.

A great deal more could be said but the most important part of this is the intentional systematic development of Godly the character. Luke 2:52 says Jesus grew in wisdom and stature and in favor with God and man.

Another list that needs to be added to this is in II Peter 1.
Grace and peace be multiplied to you in the knowledge of God and of Jesus our Lord; seeing that His divine power has granted to us everything pertaining to life and godliness, through the true knowledge of Him who called us by His own glory and excellence. For by these He has granted to us His precious and magnificent promises, so that by them you may become partakers of the divine nature, having escaped the corruption that is in the world by lust. Now for this very reason also, applying all diligence, in your faith supply moral excellence, and in your moral excellence, knowledge, and in your knowledge, self-control, and in your self-control, perseverance, and in your perseverance, godliness, and in your godliness, brotherly kindness, and in your brotherly kindness, love. For if these qualities are yours and are increasing, they render you neither useless nor unfruitful in the true knowledge of our Lord Jesus Christ. For he who lacks these qualities is blind or short-sighted, having forgotten his purification from his former sins. Therefore, brethren, be all the more diligent to make certain about His calling and choosing you; for as long as you practice these things, you will never stumble; for in this way the entrance into the eternal kingdom of our Lord and Savior Jesus Christ will be abundantly supplied to you.  Therefore, I will always be ready to remind you of these things, even though you already know them, and have been established in the truth which is present with you.  I consider it right, as long as I am in this earthly dwelling, to stir you up by way of reminder, knowing that the laying aside of my earthly dwelling is imminent, as also our Lord Jesus Christ has made clear to me. And I will also be diligent that at any time after my departure you will be able to call these things to mind. 2 Peter 1:2‭-‬15 NASB.

That is a good starting place for thinking about Biblical Character education. I welcome your comments and feedback.