They aren't interchangeable. To do one's best means that someone is trying their hardest, putting in effort. Or it could also mean to do very well. examples: "learning a language is hard, but I am doing my best to learn it". "He did his best to get a good grade on his test". To make the best of it = to find some pleasure despite difficult circumstances, to find a positive in a negative. "It was raining when they had hoped for sun. However, they made the best of it and splashed in the puddles." "Her plans were unfortunately canceled, but she made the best of it by finishing her housework" "The family made the best of a broken radio by playing fun games during the car ride."