The Rambam (Hilkhos Avel 2:3) writes that a convert is not obligated to mourn for either of his parents. This is so because someone who converts is considered as if he is reborn, and therefore has no halakhic relationship to his parents (cf. Yevamos 22a; Bava Kamma 88a). The Beis Yosef (Yoreh De'ah 374) quotes the Mordekhai in the name of the Ri that a convert must mourn for his mother, but the Rema in Shulhan Arukh (Yoreh De'ah 374:5) explicitly disagrees with this view.
But what if a convert wants to mourn his parents? I asked R. J. David Bleich and he (very briefly) said that there is no problem as long as he does not violate any prohibition, such as by tearing his clothes or refraining from learning Torah. But he certainly agreed that a convert can mourn in his own way, such as having a gathering and eulogizing the deceased. I asked my own rav, who admittedly is not a major posek, about the propriety of performing halakhic acts in non-halakhic situations, and he said that it all depends on each situation. Sometimes it might be right to sit a "fake" shivah. For example, a woman with a conditional divorce has the option to mourn over her (ex-)husband even though it is not an obligatory shivah (Rema in Even Ha-Ezer 145:9; cf. however the Pischei Teshuvah there, 7).
R. Maurice Lamm writes the following in his The Jewish Way in Death and Mourning, pp. 82-83:
There is no obligation upon a person who had converted to Judaism to mourn his non-Jewish parents in the prescribed Jewish manner. While it is expected that the convert will show utmost respect for his natural parents, he is, nonetheless, considered detached from them religiously. The grief that the convert expresses, although technically not required by Jewish law, should possess a markedly Jewish character. Therefore:UPDATE: Follow-up based on a comment, from the same book, p. 80, regrading adoptive parents:
1. The convert may say Kaddish if he so desires. It is preferable, however, since the deceased was not a Jew, for him to recite a Psalm instead, or to study a portion of Torah in honor of the deceased, as is customart on yahrtzeits. In that way, a distinction is made between mourning a Jew and a non-Jew. The decision to do either rests with the bereaved.
2. Likewise, the shiva procedures should, preferably, not be observed as in full mourning for a Jewish parent...
The converted Jew should not feel that his emotions of gried must be restrained because of religious difference. It is only the religious observance which is at issue...
There is no requirement to mourn for adoptive parents, adopted brothers and sisters, or adopted children. But while there is no legal obligation to mourn, there should be "sympathetic mourning," namely abstention from public rejoicing and similar activities in order to demonstrate a full measure of sorrow.