The Washing of the Hands
The Gemara, Sotah 39a, states that a Kohen may not perform birchat Kohanim unless he washes his hands. Rashi, ad loc., s.v. Shelo, implies that even if the Kohen washed his hands prior to the morning prayers, he must wash his hands again prior to birchat Kohanim. However, Rambam, Hilchot Tefillah U'Nesiat Kapayim 15:5, implies that if one washed his hands prior to the morning prayers, he is not required to wash his hands again immediately prior to birchat Kohanim (See Kesef Mishneh ad loc.).
Shulchan Aruch, Orach Chaim 128:6, rules that a Kohen should wash his hands immediately prior to birchat Kohanim and should not rely on his hand-washing prior to the prayer service. Nevertheless, there are still a number of situations where Rambam's opinion is relevant. First, R. Yosef Karo, Bedek HaBayit, Orach Chaim, no. 128, notes that according to Rashi, a Kohen should recite the beracha of Al Netilat Yadayim upon washing his hands for birchat Kohanim. However, since Rambam does not require him to wash his hands if he already washed prior to morning prayers, he should not recite a beracha on this second hand-washing. R. Karo codifies this statement in Shulchan Aruch, Orach Chaim 128:7. R. Eliya Shapira, Eliya Rabbah 128:13, suggests that even if the Kohen did not wash one's hands prior to the morning prayers, he should not recite the beracha. Mishna Berurah 128:24, recommends showing deference to Eliya Rabbah's opinion.
Second, Mishna Berurah 128:20, discusses a case where there is no water available for the Kohanim. He rules that if the Kohanim washed their hands prior to the morning prayers and they were careful not to place their hands in a dirty area, they may rely on the opinion of Rambam that the hand-washing prior to the morning prayers is sufficient.
Third, the Mishna, Berachot 34a, states that if the shaliach tzibbur is a Kohen, he should not perform the mitzvah of birchat Kohanim unless he is the only Kohen in the congregation and he is certain that he won't make a mistake in returning to his duties as shaliach tzibbur (see Mishna Berurah 128:76, who discusses the rule nowadays when the shaliach tzibbur uses a siddur and there is less of a concern that he will make a mistake). R. Yechiel M. Epstein, Aruch HaShulchan, Orach Chaim 128:33, rules that in a situation where the shaliach tzibbur performs birchat Kohanim, he is not required to wash his hands immediately prior to birchat Kohanim. Rather, he should rely on the opinion of Rambam that it is sufficient if he washed his hands prior to the morning prayers.
The Raising of the Hands
The Mishna, Sotah 38a, states that the Kohanim should raise their hands parallel to their shoulders in order to perform birchat Kohanim. The Midrash, Bamidbar Rabbah, 11:2, states the Almighty conveys the berachot through the gaps in the fingers of the Kohanim. For this reason, Rabbeinu Asher, Megillah 3:21, states that the Kohanim should spread their hands so that there are five gaps. Each hand has one gap between the last two fingers and the first two fingers and one gap between the first two fingers and the thumb. The fifth gap is the gap between the two thumbs. R. Yechezkel Landa, Noda B'Yehuda, Orach Chaim 1:5, rules that the raising of the hands is part of the biblical mitzvah of birchat Kohanim. He therefore rules that if a Kohen cannot physically raise his hands he does not fulfill the mitzvah. Furthermore, one cannot accomplish the raising of one's hands by leaning one's arms on something. R. Avraham Sofer, K'tav Sofer, Orach Chaim no. 13, provides a partial solution for someone who is too weak to raise his hands for the duration of birchat Kohanim. He notes that the Kohanim are only required to raise their hands for the actual recitation of the words. They may lower their hands while they are chanting between the words. R. Landa's ruling and R. Sofer's suggestion are codified by Mishna Berurah 128:52. [R. Sofer adds that if the Kohen is too weak to follow his suggestion, he should rest his arms on something, but he should not recite the beracha prior to the birchat Kohanim. Rather, he should listen to another Kohen’s beracha.]
Staring at the Hands of the Kohanim
The Gemara, Chagigah 16a, states that if one stares at the hands of the Kohanim while they are performing birchat Kohanim, his eyes become weak. The Gemara states that this only applies in the Beit HaMikdash where the Kohanim recite birchat Kohanim using the Shem HaMeforash (the explicit name of G-d). Nevertheless, the Talmud Yerushalmi, Megillah 4:8, states that even outside of the Beit HaMikdash, one should not stare at the hands of the Kohanim because it will distract one from listening to the words of the Kohanim. R. Yosef Karo, Beit Yosef, Orach Chaim no. 128, notes that for this reason, the practice developed in Egypt that the Kohanim cover their faces and their hands. They cover their faces so that they are not distracted by looking at the congregation. They cover their hands so that the congregation is not distracted by looking at their hands. This custom is cited by Rama, Orach Chaim 128:23. Magen Avraham 128:35, adds that distraction notwithstanding, one should not look at the hands of the Kohanim as a zecher L'Mikdash (a remembrance of the Beit HaMikdash). [According to Magen Avraham, it is arguable that even if the hands of the Kohanim are covered, one should not look at them because of the zecher L'Mikdash factor.]
The Face-to-Face Connection
The Gemara, Sotah 38b, states that the blessings of birchat Kohanim can only be received by those who are facing the Kohanim. Those who are behind the Kohanim do not receive the blessing. Rashi, ad loc., s.v. D'lo, explains that the blessings are only effective if the Kohanim are face-to-face with the congregation. The Gemara then proves that a person does not have to be positioned directly opposite the Kohen. It is sufficient if he is in front of the Kohanim (from their perspective) even if he is off to the side.
The Gemara also states that there is a special exception for those who cannot attend the synagogue services. Since they are in a situation beyond their control, they receive the blessings even if they are physically positioned behind the Kohanim. R. Yoel Sirkes, Bach, Orach Chaim no. 128, applies this special exception to those whose seats are located behind the Kohanim. He claims that since there is no place in the synagogue for these people to go, they are in a situation beyond their control and receive the beracha. Magen Avraham 128:36, disagrees with Bach. He notes that those people can stand at the seats of the Kohanim who are currently not occupying those seats. Furthermore, they can stand on the bimah during birchat Kohanim. Mishna Berurah 128:95, sides with Magen Avraham.
R. Hershel Schachter, Nefesh HaRav, pg. 132, notes that there are people (non-Kohanim) who turn around during birchat Kohanim in order to avoid looking at the Kohanim. He notes that this practice is not appropriate because by doing so, the congregant is no longer facing the Kohanim.