Generally speaking, Halacha forbids calling two Kohanim to the Torah for Aliyot ("Kohen Ahar Kohen"). After a Kohen has been called for the first Aliya, another Kohen should not be called, as this would give the impression that the first was not a valid Kohen. In order to avoid rumors about the first Kohen’s status, a synagogue should not call a second Kohen to the Torah.
However, if there are only Kohanim present in a Minyan, then, as the Shulhan Aruch rules, three Kohanim are called to the Torah. In such a case, it is obvious to all that additional Kohanim are called because there are no non-Kohanim, and not because of any question surrounding the status of the first Kohen. The Shulhan Aruch adds that if the Minyan consists of nine Kohanim and one Yisrael, then the Yisrael should be called to the Torah for the first Aliya, out of concern for "Darcheh Shalom" (peaceful relations among the congregants). The Mishna Berura (commentary by Rav Yisrael Meir Kagan of Radin, 1839-1933) explains that if one of the nine Kohanim would be called up first, the other Kohanim might feel resentful and envious. Therefore, the Shulhan Aruch writes that the Yisrael should be called first, and then Kohanim are called for the subsequent Aliyot, to avoid contention.
The Rama (Rabbi Moshe Isserles of Cracow, 1525-1572), in his glosses to the Shulhan Aruch, writes that these Halachot apply to Leviyim, as well. If a Minyan consists entirely of Leviyim, then three Leviyim are called – even though we generally do not call a second Levi after a first Levi has received an Aliya – and if there are nine Leviyim and one Yisrael, then the Yisrael is given the first Aliya and Leviyim the next two.
The Mishna Berura, in this context, addresses the case where the only Kohen in the synagogue is a mourner within the first week of mourning, Heaven forbid. If this occurs on a weekday, he writes, then the Kohen may not be called for the first Aliya, because a mourner is forbidden from studying Torah. A Yisrael is therefore called to the Torah in his stead. On Shabbat, however, he is called for the first Aliya. If he would not be called for the first Aliya, this would constitute a public demonstration of mourning, which is forbidden on Shabbat. Therefore, if a mourner is the only Kohen in the synagogue on Shabbat, he should be called for the first Aliya. However, the Mishna Berura adds that it is preferable for him to exit the synagogue before the Torah reading, so that a Yisrael can be given the first Aliya instead of the mourner.
Summary: Generally, only one Kohen is called to the Torah for every Torah reading, but if everyone in the synagogue is a Kohen, then all the Aliyot are given to Kohanim. If there are nine Kohanim and one Yisrael, then the Yisrael is called for the first Aliya, and Kohanim are called for the subsequent Aliyot. These laws apply to Leviyim, as well. If a mourner is the only Kohen in the synagogue, on weekdays he should not be given an Aliya, whereas on Shabbat he may be called for the first Aliya, though it is preferable for him to leave the synagogue before Torah reading so that the first Aliya can be given to a Yisrael.
