I have already performed the obligation of Hajj. My grandmother 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.

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, 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 dead or physically unable and they do not expect recovery. The Prophet, sallallaahu ‘alayhi wa sallam, 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 asked you to perform Hajj on her behalf, it is permissible for you to do so. Though, 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.

But if your father is physically unable to perform Hajj and he asked 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 that a man said: "O Messenger of Allaah, who amongst the people is most deserving of my good treatment?" He said: "Your mother, again your mother, again 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, sallallaahu ‘alayhi wa sallam, 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 did not ask you to perform Hajj on his behalf, then you can perform Hajj on behalf of your grandmother and you are rewarded for this by the Will of Allaah.

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, you intend Hajj on her behalf, assume Ihraam from the Meeqaat and say Talbiyah on her behalf and then complete the other rituals of Hajj in the ordinary manner.

The reward of this Hajj will reach her and benefit her Inshaa’ Allaah and absolve her of liability to perform the obligatory Hajj.

You are not required to do anything more than this.