When a non-Jew undergoes conversion and becomes a Jew, he is, halachically speaking, no longer related to his biological parents. Nevertheless, if a convert's parent passes away, it is appropriate for him to recite Kaddish to benefit the parent's soul. He of course owes his parents a debt of gratitude for bringing him into the world and caring for him, and so it is proper recite Kaddish to benefit their soul. Although the parents are not Jewish, nevertheless, the Kaddish recitation enables their souls to rest peacefully in the afterlife, and it spares them from punishment.
This is the ruling of Hacham Ovadia Yosef, who noted the discussion of Tosafot (Sota 10b) regarding King David's prayers for the soul of his son, Abshalom. Tosafot establish that although a person's Misvot cannot benefit the soul of his deceased child (the way a person's Misvot benefit the soul of a deceased parent), a person's prayers can benefit his deceased child's soul. And thus King David's prayers to elevate his son's soul from Gehinam were effective. Hacham Ovadia applied this principle to the case of a convert. Since Halacha does not regard the convert as the biological parent's child, his Misvot cannot benefit the parent's soul, but his prayers can benefit the parent's soul. Therefore, it would be appropriate for the convert to recite Kaddish for the deceased parent.
An adopted child should recite Kaddish for the parents who adopted him and raised him. He of course owes them an enormous debt of gratitude for all they did for him, raising him, caring for him, and assuming responsibility for him, and so he should recite Kaddish after their passing to benefit their souls.
Summary: It is appropriate for a convert to recite Kaddish for his deceased parent. Likewise, a person who was adopted should recite Kaddish for the parents who adopted him and cared for him.