That is a very nice question and one which needs a lot of brainstorming. And definitely not to leave the heart behind. The character of a man is how we perceive him in what situation. In general day to day life being a man of character or woman of character has been confined to sexual morality that has been portrayed by the society - living with one partner, not cheating on your partner, not flirting around or using loose remarks against opposite sex etc. "Character" has been narrowed down to that.
But for me character is vast and in it's entirety includes everything that makes you a good human and it is irrespective of gender. A widow seeking comfort and support and entering into second marriage is not characterless if she has not lured or bargained the man into a forced relationship. It is a high demand in our society to accept people as adult and respect their choice of life even if it is in contrast to our choice. As long as it does not affect the harmony of society, it should not be questioned.
If we look into History, Adolf Hitler was loyal to his wife. But was he a man of character?? Einstein had multiple relations with various ladies but still he had immense contribution to the mankind.
An alcoholic damages his health but can be kind and generous to society. A person who fits in to conventional picture of strong character- disciplined, religious...etc.etc. might not hesitate to compromise with ethics for his hunger for power and money.
Evaluating a character needs a wider lens. Even Lord Ram's character as a husband can be questioned as he abandoned his innocent wife just for some blame made by an unknown dhobi. He despite of being a man of strong character failed as a husband. On the contrary, Ravana despite of being an evil does scores a few plus point for being a gentleman who respected Sita's modesty and did not touch her.