Question
I have already performed the obligation of Hajj. My grandmother has asked me to perform Hajj on her behalf because she is physically and financially unable to do so. Is it permissible to perform Hajj on her behalf? What are the conditions that I am required to fulfill towards her? Please note that my father is also physically unable to perform Hajj.
Fatwa
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.
If you have already performed Hajj for yourself, it is permissible for you to perform Hajj on behalf of those who are deceased or physically unable and who do not expect to recover. The Prophet said to a person who was performing Hajj on behalf of another: "Perform Hajj on your own behalf, then perform it on behalf of Shubrumah." [Abu Daawood and others]
If your grandmother is physically unable to perform Hajj and recovery is not expected and she has asked you to perform Hajj on her behalf, it is permissible for you to do so. This is despite the fact that she is not required to perform Hajj because she is financially unable, as you mentioned. You should do so out of perfect dutifulness to her.
However, if your father is physically unable to perform Hajj and he asks you to perform Hajj on his behalf, his request takes precedence because he is worthier of your dutifulness. It was narrated on the authority of Abu Hurayrah, may Allah be pleased with him, that a man said: "O Messenger of Allah, who amongst the people is most deserving of my good treatment?" He said: "Your mother, then your mother, then your mother, then your father, then your nearest relatives in order." [Muslim]
An-Nawawi said,
Al-Qaadhi said, "The scholars unanimously agree that the mother and father are worthier of their child's dutifulness than others. But they (the scholars) provided different views regarding grandfathers and siblings, as the Prophet said: “Then your nearest relatives in order.” The Shaafi‘i scholars said: "It is recommended to give precedence in dutifulness to the mother, then the father, then the children, then the grandfathers and grandmothers."
But if your father has not asked you to perform Hajj on his behalf, then you can perform Hajj on behalf of your grandmother and you will be rewarded for this, if Allah wills.
As for your question about the conditions that your are required to fulfill towards your grandmother, if you mean the manner of Hajj, as it seems from your question that you intend Hajj on her behalf, then it is to assume Ihraam (ritual consecration) from the Meeqaat (designated point for assuming Ihraam) and say the Talbiyah (Hajj-related initiation chant) on her behalf and then complete the other rituals of Hajj in the standard manner.
The reward of this Hajj will reach her and benefit her, if Allah wills, and absolve her of the liability to perform the obligatory Hajj.
You are not required to do anything more than this.
And Allah knows best.