The Shulhan Aruch (Orah Haim 128:20; listen to audio recording for precise citation) addresses the question of whether or not a Kohen who serves as Hazzan should recite Birkat Kohanim (the priestly blessing). If other Kohanim are present in the synagogue, the Shulhan Aruch writes, then the Kohen should not participate in the recitation of Birkat Kohanim. The Rama (Rabbi Moshe Isserles, Poland, 152o-1572), in his glosses to the Shulhan Aruch, adds that in this case, nobody should call the Hazzan for Birkat Kohanim, as they call the other Kohanim. It is forbidden for a Kohen to refuse an invitation to recite Birkat Kohanim, and therefore in this case, when Halacha advises the Kohen not to join the other Kohanim in the Beracha, one should not call him for the blessing (such as by saying, "Be'chabod" and the like).
The Shulhan Aruch then rules that if no other Kohanim are present in the synagogue, then the Hazzan has the option of reciting Birkat Kohanim. If he feels capable of reciting the Beracha and then returning to the repetition of the Amida without becoming flustered, then he may recite Birkat Kohanim. The Sages allowed him to recite Birkat Kohanim in this case in order that the congregation will not miss an opportunity to receive the Beracha. The Mishna Berura (commentary by Rabbi Yisrael Meir Kagan, 1839-1933) adds that nowadays, when everyone has access to printed Siddurim, it is generally assumed that a Hazzan can easily maintain his composure and will not become flustered after reciting Birkat Kohanim. Nevertheless, if a Kohen feels, for whatever reason, that it would be too confusing to recite Birkat Kohanim and then return to the Amida, then he should not do so.
When a Hazzan recites Birkat Kohanim, he must move his feet two steps or so towards the Duchan (area when the Kohanim recite Birkat Kohanim) during the Beracha of "Reseh." Just as all Kohanim in the synagogue are required to make an "Akira" – meaning, to leave their place – when the Hazzan reaches "Reseh," in preparing for the Beracha, so must the Hazzan move his feet slightly if he will be reciting Birkat Kohanim. Even though he is in the middle of reciting the repetition of the Amida, he must nevertheless take a few small steps forward. What more, the Shulhan Aruch writes that the Kohen should actually move to the front of the synagogue for Birkat Kohanim in this case. However, if this would cause the Hazzan confusion, or if he might begin speaking as a result of moving from his place, then he may recite Birkat Kohanim in his place (but he must nevertheless make an "Akira"). In such a case, those sitting toward the front of the synagogue must ensure not to stand in front of the Hazzan.
When the Hazzan recites Birkat Kohanim, somebody else calls out the words of the Beracha. The Shulhan Aruch maintains that if this situation was foreseen in advance, then the person assigned to call out the words of Birkat Kohanim should listen very attentively to the Hazzan's repetition, and then complete the repetition in the Hazzan's place after Birkat Kohanim. This is not, however, the practice nowadays. Rather, the Hazzan himself completes the repetition after Birkat Kohanim.
If the Hazzan in this situation chooses not to recite Birkat Kohanim, then somebody else from the congregation should recite the paragraph of "Elokenu V'Elokeh Abotenu Barechenu…," since a Kohen is not permitted to recite this paragraph. This ruling is mentioned by Hacham David Yosef, in his work Halacha Berura.
In general, it is preferable for a Kohen not to serve as Hazzan. Birkat Kohanim fulfills a Torah obligation, and is thus afforded greater importance than serving as a Hazzan, which, though certainly very admirable, does not have the status of a Torah-ordained Misva. A Kohen should therefore not serve as Hazzan, which might cause him to miss the opportunity to fulfill the Misva of Birkat Kohanim. There is more room to allow a Kohen to serve as Hazzan on Shabbat, since Birkat Kohanim is recited at both Shaharit and Musaf, so he will still have the opportunity to recite the Beracha once that day. During the week, however, it is preferable to choose a non-Kohen to serve as Hazzan.
Summary: A Kohen who serves as Hazzan does not join the other Kohanim for Birkat Kohanim. For this reason, it is preferable for a Kohen not to serve as Hazzan. If there are no other Kohanim in the synagogue, then the Hazzan does recite Birkat Kohanim, unless he is concerned that this will undermine his ability to focus on properly reciting the repetition of the Amida.