Singing alone does not convey the meaning.
Although I have read Smantha's further comment, I recommend that the word "out" should be in there.
Alternatively, "I unburden myself to you" would be a way one could say it in English. Another way would be to say, "I reveal my innermost feelings" to you. Although one could say in English, "He sings his heart", I feel that one would be unlikely to say, in the first person, "I sing my heart to you" without the word "out".