A Godly character is more than just trying to do the “right” thing. Trying puts the emphasis on our own effort, which puts ourselves first. Jesus pointed out that focusing on our efforts to keep the whole law can lead us to trip over small stuff and miss the big picture. The “big picture” has its own traps, however.
Some years back there was a lot of talk about “situation ethics”, which sought to justify exceptions to traditional values. There are times that the context is critical, but often it's an excuse for actually dismissing the Biblical ethics of the situation, arrogantly thinking we know more than God. What seems to be right and helpful in the short run can be damaging in the long run.
“Character” is one of those things that may be tricky to define, but we “recognize it when we see it.” I have noticed, in watching people over the years, that high ideals and a good reputation in one area of life may not be consistent with other areas. For example, we have seen many major public figures with great concern with making society better, fairer, etc., believe that their approach and their goals are all important, and any alternates or questions are inadmissible. Often, a good public image covers up shortcomings in everyday personal life. Occasionally an admirable private life is an excuse for evil and selfish public behavior. We're not surprised to find that people often don't have all sides of their lives in order; somehow it can make us feel better about our own shortcomings—but it's very difficult to live down the damage to a relationship. Ultimately, we can only maintain Godly character not by our own strength, but through the grace and mercy of God.
* Proverbs 16:25