One of the more complex relationships is that of a convert to their biological parents. On the one hand they are the child’s parents and comes with that the love and caring that any child has for a parent. On the other hand, after the conversion the child is not halachically related to the parent’s anymore. This creates many halachic situations as we shall discuss throughout this sefer. One specific area of discussion is how the child relates to the parents immediately following their passing, more specifically related to reciting kaddish. The question is whether the child can recite kaddish for his biological parent or is this to be avoided. Indeed, the poskim maintain that a convert can pray for the health of an ailing parent. (Refer to Umekareiv Biyamin vol. 1 Siman 5) What needs to be clarified is whether kaddish is different. 

Section 1: The Views Of The Poskim
Rav Ahron Walkin zt”l (Zaken Ahron 2 Y.D. 87) was asked this very question and he writes simply that while he is not obligated to do so, it is definitely permitted to do so. He basis this on the teaching of the Rama that one may recite kaddish for a Jewish renegade who rebelled against Hashem. If one can recite kaddish for a Jew who actively sinned, all the more so that we can recite kaddish for a non-Jew who did not actively revolt against Judaism, as he is not Jewish. Rav Ovadia Yosef zt”l (Yechave Daas 6:60) likewise rules that it is permitted for him to recite kaddish.
Rav Walkin notes that we tend to avoid to offer halachic rulings that seem strange to the Orthodox public, such as reciting kaddish for a non-Jew. Therefore, if he finds himself around people that knows that he is a convert and that his father was not Jewish it is better for him to recite Tehillim for his father’s soul, as opposed to reciting kaddish. If, however, he is in a location where they do not know his familial background, he may recite Kaddish.
Rav Yaakov Hillel shlit”a (Vayeishev Hayam 3:38) discusses this question at length. He maintains that since the purpose of kaddish is to alleviate the soul’s suffering in Gehinnom and in order to aid the soul’s elevation it is not something that is appropriate to be recited for a non-Jew. The soul of a Jew differs from that of a non-Jew and the prayer of kaddish will ultimately help that of a Jewish soul and not that of a non-Jew.109

Summary and Conclusion
The poskim debate whether a convert may recite kaddish for his biological parent. According to Rav Ahron Walkin zt”l while he is not obligated to do so, it is definitely permitted to do so. Rav Walkin notes that we tend to avoid to offer halachic rulings that seem strange to the Orthodox public, such as reciting kaddish for a non-Jew. Therefore, if he finds himself around people that knows that he is a convert and that his father was not Jewish it is better for him to recite Tehillim for his father’s soul, as opposed to reciting kaddish. If, however, he is in a location where they do not know his familial background, he may recite Kaddish. This is also the view of Rav Ovadia Yosef zt”l. However, Rav Yaakov Hillel shlit”a opposes the idea of a convert reciting kaddish for his non-Jewish parent. He maintains that since the purpose of kaddish is to alleviate the soul’s suffering in Gehinnom and in order to aid the soul’s elevation it is not something that is appropriate to be recited for a non-Jew. For normative halacha, a rabbi should be consulted.