The passage from Matthew 6:4 seems mostly contradictory from a chapter earlier in Matthew 5:16:
“Let your light shine before men, that they may see your good deeds and praise your Father in Heaven.”
Jesus wants the world to see how Christians live their lives: living as “salt and light” and everyway that he outlines (Matt. 5-7 is a good start). As you said, a good deed may “be considered most genuine when done in privacy”, but I don’t think that Christians are to do all of their good deeds in secret.
Therefore, Christians – like Bieber – can and should give publicly; yet, (t)he(y) should only do so in a way that isn’t making a performance out of it.
But, “Is there any act of charity that Justin Bieber can do without making it a performance?” Certainly: Justin Bieber could go to a hospital and visit sick kids as part of a contribution to the “Make a Wish Foundation”. (As seen in the “Pray” video.)
The line begins to blur when he is seen doing charity work in his music videos.
You can choose to see Bieber as promoting ‘caring for the sick’, raising awareness for ‘social issues’, and a promoter of doing good things – being a good Christian. Bringing camera’s wherever he goes is a way for spreading his ‘light to the world’.
Oppositely, you can choose to see him as a self-promoting, image-making superstar, combining messages of inequality and his acts of charity to conger up people’s emotions and ‘fall in love’ with him. Bringing camera’s wherever he goes is a way of making a performance out of it.
Is a public act of charity insincere?
Or, for the Biebester (or any famous person), is it the fact that he’s making (possibly more) money off (the footage of) his acts of charity that makes him insincere?