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?