Why should parents to pay attention to their own character, first and foremost?
Most parents put at least a little effort into giving their kids some sort of character education. Children learn from consistency and repetition, both of what they hear and see. Anything that children learn comes from proximal development (being around what they learn) - theorist was Albert Bandura - and parental modeling of the behavior gives many years of seeing and hearing the character traits in action. Until children reach adolescence (and sometimes not even then), they are concrete and literal learners. They will not understand abstractions, such as "responsibility", "character", "integrity", but they will understand "doing the right thing" which means applying all those character education traits.
Children get to see those traits in many applications, for example honesty is in more than telling the truth as it is making sure you are charged the correct amount (more/less than shown on the register), getting the correct change back after a purchase (giving extra back), returning found items to the person you know they belong to, and dozens of other applications. The more often parents display the character traits in all settings (home, community, extended family, workplace, etc.), the more examples children have to emulate.
How can parents teach kids about character through their own actions?
Parents need to be conscious of what their character trait/values are and make sure they are consistent. It does not matter what the economic conditions are (easy money times may actually be harder to teach many of the values because it is too easy to buy one's way out of/through situations). In many ways our society has lost the work ethic (diligence, honesty, responsibility, dependability, pride in task accomplishment/completion, etc.). Perhaps these times will see a marked resurgence of demands for a work ethic in employees. If people want to have a job, they have to demonstrate qualities employers value, not what the employees are willing to do. When parents lose jobs because of their irresponsible patterns, children may learn; a better teacher would be for parents to demonstrated qualities employers want to keep.
What specific steps can adults improve ourselves if needed?
Adults need to determine what character traits are valuable (important) to them. They then need to determine how those traits appear to others. The hard part is being honest with self in a self-evaluation to determine if the individual consistently demonstrates those traits. The individual may need to become aware of how (s)he sabotages his/her intent by acting/speaking unconsciously or choosing a short-cut to accomplish a goal. The individual must determine if it is worthwhile to consistently demonstrate the desired traits; if so, then (s)he needs to act/speak according to the standards (s)he sets for the self and children (s)he is trying to teach. This process is akin to the moral inventory step of any 12 step program. It must be applied daily for it to work.
What about the saying 'do as I say, not as I do' when it comes to raising kids with solid character?
Hypocrisy is not a valued character trait. It is like lying. No one likes being lied to. The words of saying are not sufficient to wipe away the many visual opportunities to witness the adult doing the "forbidden" behavior. Kids will ignore the hypocrite and determine their own values (most likely not what the adult is advocating verbally)
