If my father passes away with no Will and I'm currently living in his home, since I am the eldest child what are my rights?

Assuming you have another sibling or more, and no will in existence, then a petition for administration of the estate will need to be filed for distribution of the estate in equal shares, and the estate/remaining siblings are entitled to rent from you while you occupy the same. Merely because you are the oldest does not give you any preference in the estate distribution.