Each family may approach "character" differently based on tradition, based on culture, based on experience and based on knowledge acquired. One of the BEST things you can do for your family is WRITE DOWN ultimate goals your family should have in order to create listening ears, attentive eyes and happy hearts.
Some Character topics are discussed in NON-CHRISTIAN homes and are effectively applied. It does not mean that Christianity and Christ's eternal presence is not needed in those homes, it just means that the general principles are taught. Some NON-CHRISTIAN homes apply character principles for children better than Christian homes due to experience of suffering or other matters.
Here is an example of general Character rules that can be applied to any home, Christian and Non-Christian.
1. Use soft and kind words even if you can't solve a problem. Kindly ask for help.
2. Find many opportunities to smile and be thankful for others.
3. Find ways to praise other's godly character, achievements and accomplishments.
4. Always use manners and be respectful of others and their belongings.
5. Always do right, even when others may not and when no one is looking.
6. Practice using listening/ empathy skills and understand how others feel. Allow Holy Spirit to correct those in error along with # 24.
7. Be thankful for how you were created by God. Find ways to honor him by obeying his word in that reguard.
8. Never mock, tease or put others down.
9. Never argue, complain or blame others for their behaviour. Pray for them in secret.
10. Speak to an accountability prayer/partner (spouse or brother/sister).
11. Do not boss or try to take control of others or try to meddle into their business and privacy for selfish gain.
12. Clearly communicate how you have been hurt using the word "I" never "you" and stating the problem with kind words and actions. Along with #9. Approach anger in a kind manner.
13. Forgive immediately.
14. Look for ways to serve others with no though of self gain.
15. Think pure thoughts about self and others. Put no stumbling block before those who are weak.
16. Put no wicked thing before your eyes. Psa 101:3
17. Never, gossip, tale-bear or listen agreeably to those who speak reproachfully of others. Only report truthful dangers.
18. Never raise a hand to hit.
19. Never raise a foot to kick.
20. Never raise an object to throw.
21. Never raise a voice to yell.
22. Never raise an eye to scowl.
23. Never use body or objects to harm or injure others in any way (pushing, spitting, bonking etc)
24. Never lift up nose to think pridefully of self as better than others in any area or subject.
25. Share toys/objects/activities. Use or do one activity at a time.
26. Do your best to keep surroundings neat, clean and organized. Clean up after your own mess.
27. Always approach daily tasks in a quiet peaceable manner without grumbling or complaining. Organize and do things orderly.
28. Never let the sun go down on your wrath. Don't go to bed angry or guilty.
29. Jesus first, other's second, yourself last.
Review regularly from books that teach character. For example: https://www.amazon.com/Family-Virtues-Guide-Publisher-Plume/dp/B004UJM4Q0/ref=sr_1_2?ie=UTF8&qid=1501879152&sr=8-2&keywords=The+Family+Virtues+Guide
Review regularly Character stories and traits from the bible. For example: Psalms and Proverbs and http://www.iblp.org/programs/biblical-character or illustrations from www.sonlighteducation.com
Study cultures and how they applied Character to their culture: http://ojibweresources.weebly.com/ojibwe-teachings--the-7-grandfathers.html