Is it permissible to give zakat to a poor brother whose salary is insufficient to cover his children's expenses and their private school fees?

If your brother finds enough for himself and those whose expenses he is responsible for from basic needs, then it is not permissible to give him from zakat money. Educating children in private schools is not among the needs for which zakat is paid. As for helping your brother with charity money to provide more for him and his family in permissible matters, it is a good deed, and charity to a relative is more meritorious.