How do you teach them about integrity?

I was watching a video that Jezuzfreek777 posted on his YouTube account. He was taking his daughter to a park, and while she was playing, he was preaching. I watched the video, and it got me to thinking about integrity.

One of the main things I learned in the Army was to have integrity. To do the right thing all the time, not just when you know someone is watching. Now I know no one can have a perfect record of integrity, but in my opinion that's what people should strive for. Being as honest as possible to the highest degree that you can.

Anyhow, I was wondering how a religious person could teach a child both about God and integrity, seeing how a believer is never without at least one being watching them.