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If a Jewish man marries a non-Jewish woman, their children are not considered to be Jewish. (Some Jews don't, or didn't, agree). Presumably, if the man was a Kohen, any son he had would not be a Kohen because the son wouldn't be Jewish.
If the son converts, could he be a Kohen? I assume that the father, although disqualified from service because of intermarriage, never stopped actually being a Kohen. Could the son claim his patrilineal heritage upon becoming a Jew?
There is no chain of descent. This is similar to the questions about relatives who convert and the various laws of inheritence. A man who has a child by a non-Jewish woman is not considered to have fulfilled the mitzvah of Pru U'Rvu and the child has no connection with him. This is analogous in the case of the person who blasphemed in the desert. The meforshim (Ramban, Rashi, Medrash) explain that he was the son of a Jewish woman (and so Jewish) but because his father was a nonJew he could not be a member of the tribe of Dan. A convert cannot be a member of a tribe (no matter who his father was) and this would also apply to being a 'descendant of Aharon'.
Another example is that the Torah talks of someone who dies with 'no heirs'. Only a convert can be in this situation because even if he and his children (or other relatives) convert, they are not considered relatives. It is only a rabbinic prohibition that forbids marriage between converts who are relatives that would otherwise be considered incest.
