How do you determine just who are a deceased person's legal heirs?

Sorry to hear about your brother and your parents.

Since your brother did not have a Will, the rules of intestate succession determine inheritance.  Intestate means dying without a Will.  Under intestate succession, your brother's estate would have gone to his spouse, but since there is no surviving spouse, his estate will be divided equally among his surviving children.  The fact that they are step-children does NOT change this as they inherit the same as biological children.  If there had not been any surviving children, your brother's estate would have been divided equally among his surviving siblings.