Is it permissible for me to perform Umrah on behalf of myself and my deceased father, and a number of other deceased relatives? What is the permissible number and what are the conditions for that? And is it a condition to make that intention before traveling?

If you intend to enter into ihram for one ritual for yourself and for others, it is not permissible, because ihram can only be consecrated for one person. However, it is permissible to share the reward of the umrah with others, and Sheikh Al-Uthaymeen was of the opinion that there is no limit to sharing in the reward.

It is better that you dedicate the reward of your umrah to yourself and strive in supplication for the deceased. If you wish to perform umrah on their behalf, you can go out to the nearest point outside the haram boundaries (adna al-hill) after completing your umrah, and then enter into ihram for umrah on behalf of whomever you wish, repeating this as many times as you desire during the same journey.

There is no condition to make this intention before traveling.