When Obama refers to our character whose character is he referring to?

While I do not know which speech you are referring to, President Obama (along with other presidents, and many public speakers in leadership roles) frequently uses what is called the "royal we" when he speaks. This is a common rhetorical device-- to use "we" and "our" rather than "I" and "my" when giving certain speeches. While the president is speaking as the leader of the nation, he also wants all Americans to be connected to what he is saying. So, he might talk about "our character"-- meaning the character (the virtues, the ethics) of all of us.