Before the Kohanim raise their hands to bless the nation, they must wash their hands up to their wrists. There is an allusion to this in the verse (Psalms 134:2), “Lift up your hands in holiness and bless Hashem.” Any Kohen who did not wash his hands may not bless the people (Sotah 39a). Eminent Rishonim disagree regarding the reason for this hand washing. According to the Rambam (Tefillah 15:5), the obligation to wash one’s hands only exists when they are dirty. Therefore, a Kohen who washes his hands in the morning and is careful not to touch dirty parts of his body is not obligated to wash his hands again before Birkat Kohanim. However, according to Rashi and Tosafot (Sotah 39a), even clean hands must be washed and sanctified in honor of Birkat Kohanim.
In practice, we are stringent, and a Kohen whose hands are clean must also wash them before Birkat Kohanim, like the opinion of Rashi and Tosafot. However, no berachah is recited on the washing even if he touched dirty places before washing, since there is doubt whether or not the Chachamim instituted a berachah for it (Mishnah Berurah 128:24, based on the Eliyah Rabbah).
It seems from the Zohar that this washing was intended to sanctify the Kohanim, and their hands, in preparation for the berachah. In order to increase the level of sanctity, it is customary that a Levi pour the water on the hands of the Kohanim. When there is no Levi present in the synagogue, it is advisable that a firstborn, who possesses some amount of sanctity, wash the hands of the Kohanim. In the absence of a Levi and a firstborn, the Kohen washes his own hands (Shulchan Aruch 128:6; Mishnah Berurah 22).
When the Kohen is concerned that perhaps there will not be water in the synagogue with which to wash his hands before blessing the people, he washes them in his house before the prayer service and takes care to keep them clean. In that way he can recite the blessing upon the people. However, if the Kohen is not careful to keep his hands clean after the morning washing, and he does not have water to wash them before Birkat Kohanim, he may not ascend the duchan to bless the nation.
In every situation where the Kohen is forbidden to bless the people, it is advisable that he leave the synagogue before the time of Birkat Kohanim, so that the gabbai will not mistakenly call him up to ascend the duchan when he is not permitted to do so. If the Kohen remains present and he is not called, there is concern that perhaps people will think that he is an invalid Kohen.