"Content" in this context means "Contents"; in other words, what is "held within" or, simply "inside".
"Character" is what makes someone what he (or she) is. It includes what they believe in, but also - and most importantly) -how they behave: for example, whether or not they are honest, kind, friendly, loving, respectful and so on. It is similar to "personality", but tends to be used when considering whether someone is good or bad, and focuses on their actions. So, one could have a violent gangster who has a thoroughly bad character, but a "lively" personality!
So, Martin Luther King was really asking that people be judged by their behaviour rather than their appearance.