First of all, we would like to clarify that it is permissible to give zakah to relatives such as brothers, sisters, uncles, aunts, nephews, and nieces if they are poor and in need of help. However, zakah cannot be given to a wife, children, grandchildren, parents and grandparents as it is one’s duty to take care of their needs.
Also, it is not permissible to give zakah to one’s relative in order to help him perform Hajj.
In responding question related to this, Sheikh Ahmad Kutty, an Islamic scholar stated:
Although you are allowed to give your zakah to your poor relative—provided he is not one of your dependents—you are not allowed to give it to anyone for the specific purpose of performing Hajj. The reason for this is that Hajj is only obligatory upon those who are financially able to do it. Almighty Allah says: [And pilgrimage to the House is a duty unto Allah for mankind, for him who can find a way thither.] (Aal `Imran 3: 97) According to scholars, financial ability is one of the most important factors to consider in determining the obligation to do Hajj; if not the only one.
The recipients of zakah have been mentioned in the Qur’an as follows, [The (obligatory) alms are only for the poor and the needy, and those who collect them, and those whose hearts are to be reconciled, and to free the captives and the debtors, and for the cause of Allah, and (for) the wayfarer; a duty imposed by Allah. Allah is Knower, Wise.] (At-Tawbah 9: 60)
Hence, zakah is to be given to those who are more deserving than those who wish to perform Hajj, which is not obligatory upon them.
In conclusion, you are not allowed to give zakah to your relative to perform Hajj, especially when there are far more deserving people to receive it.