Perhaps character could be described as our reputation with ourselves. In that sense, we can strive to be the kinds of people that we ourselves hold in high regard, improving our internal character/reputation/image.
Let’s not pretend that it doesn’t matter what others think about us. Of course it matters. Yet it’s still possible to prioritize character over reputation. In the end, both your character and your self-worth is determined by you, not what others might say about you.