In this week's Parsha, the Torah articulates the Beracha which the Kohanim are supposed to pronounce when blessing the Jewish people (במדבר ו:כ"ד-כ"ו). The Gemara in Kesubos (:דף כ"ד) states that if a Jew who is not a Kohein raises his hand and recites this Beracha, known as the Birchas Kohanim, he violates a Mitzvas Asei by so doing. Rashi (שם בד"ה דאיסור) explains this by citing the Posuk in our Parsha that introduces the Birchas Kohanim (שם פסוק כ"ג), which says, addressing the Kohanim, כה תברכו את בני ישראל, so shall you bless Bnai Yisrael, from which the Gemara derives, according to Rashi, אתם ולא זרים, only you, the Kohanim, may pronounce this Beracha and not any non-Kohein. Such a derivation is called a לאו הבא מכלל עשה, which means that one violates a Torah law by ignoring this Mitzvah, although it is expressed in the positive form. It seems clear from here that a non-Kohein may not bless his child, or a student, or a friend, using the Pesukim of Birchas Kohanim.
The Gemara in Shabbos (דף קי"ח:), however, records that one of the Tannaim stated that although he was not a Kohein, he would, upon the request of his colleagues, go up to the דוכן, the platform, and, presumably, recite Birchas Kohanim anyway. In Tosafos there (שם בד"ה אילו), the Ri is quoted as saying that he does not know of any problem with a non-Kohein going up to Duchen, except the problem of a ברכה לבטלה, an inappropriate Beracha, which would be recited, because really only the Kohanim were commanded to bless Bnai Yisrael. The implication of this statement is that it would be permissible for a non-Kohein to recite the words of the Birchas Kohanim and bless someone as long as he would not precede this recitation with a Beracha. The Ramo, in his commentary on the Tur entitled Darkei Moshe (אורח חיים סימן קכ"ח אות א'), quotes a view that based upon this, it would be permissible for any non-Kohein to go up to Duchen along with the Kohanim, and that it may in fact be proper because of the idea that ברוב עם הדרת מלך, meaning that it represents greater glory for Hashem to have more people participating in a Mitzvah. He notes, however, that this is not the practice, perhaps because the non-Kohanim may get confused and eventually go up to Duchen even when no Kohanim are present.
The Ramo (שם) then asks, however, how Tosafos can allow this at all, and say that there is no problem for a non-Kohein to go Duchen, when the aforementioned Gemara in Kesubos (שם) implies that a non-Kohein in fact violates a Mitzvas Asei if he Duchens. The Ramo tries to reconcile the two sources by suggesting that the Ri's rule permitting a non-Kohein to Duchen is intended only if he joins the Kohanim when they Duchen, whereas the Gemara in Kesubos is discussing a non-Kohein who Duchens when no Kohanim are there; in such a case, the non-Kohein indeed violates a Mitzvas Asei. Despite his answer, though, the Ramo still leaves the matter as a question to be pondered. The Maharsha in Shabbos, in commenting on the above cited Tosafos (חידושי הלכות לשבת שם), suggests that it is indeed forbidden for a non-Kohein to recite the words of the Birchas Kohanim, as the Gemara in Kesubos (שם) indicates, and that the Tanna's comment that he would go up to Duchen, despite not being a Kohein, meant that he would go up to join the Kohanim when they would Duchen, but he would not say the words of the Beracha. It is this practice which the Ri in Tosafos (שם) found acceptable. But a non-Kohein may certainly not pronounce the words of the Birchas Kohanim.
In the Shulchan Aruch (או"ח שם סעיף א'), the Ramo rules that a non-Kohein should not raise his hands to Duchen, even with other Kohanim present. The Magen Avraham (שם ס"ק א'), in his attempt to reconcile the rule of the Gemara in Kesubos (שם) that a non-Kohein who goes to Duchen violates a Mitzvas Asei with the practice of the Tanna in the Gemara in Shabbos (שם) sanctioned by Tosafos (שם), claims that the Mitzvas Asei violated is the inappropriate pronunciation of Hashem's name in the Birchas Kohanim, and that the Tanna who went up to Duchen only went up, but did not say anything. The Noda BiYehudah (שו"ת נודע ביהודה מהדורא קמא חלק או"ח שאלה ו') presents two ways to understand what the Magen Avraham meant by an inappropriate use of Hashem's name; either he was referring to Hashem's name in the Beracha which is recited prior to the actual Birchas Kohanim, or to Hashem's names which appear in the actual Pesukim of the Birchas Kohanim. He seems to prefer the latter explanation, implying that even saying the Pesukim is a problem for a non-Kohein; Rav Ovadyah Yosef (שו"ת יביע אומר חלק ג',חאו"ח סימן י"ד אות ח'), however, thinks there is no problem for anyone to say Pesukim from the Torah, even with Hashem's name in them, and that the Magen Avraham was rather concerned about mentioning Hashem's name in the Beracha preceding the actual Birchas Kohanim, a view which he quotes others as holding as well.
The Noda BiYehudah (שם) also suggests a distinction between Birchas Kohanim when it was recited in the Beis HaMikdash as opposed to the Birchas Kohanim recited today, saying that perhaps only in the Beis HaMikdash was it forbidden for a non-Kohein to recite this Beracha, while nowadays it is a Mitzvah only upon Kohanim, but a non-Kohein may recite the Birchas Kohanim if he wishes. The Minchas Chinuch (מצוה שע"ט אות א') also quotes such an interpretation, and the Pnei Yehoshua In Kesubos (שם בד"ה בגמרא) suggests this as well, noting that in the Beis HaMikdash, the full name of Hashem was used, and this could not be said in the Beracha by a non-Kohein. It appears from at least some of the above sources that it would be permissible for a non-Kohein to bless someone using the words of the Birchas Kohanim nowadays, especially if he does not recite any Beracha before doing so. The Torah Temimah on our Parsha (במדבר ו; כ"ג, אות קל"א), though, suggests that even a Kohein may not be permitted to bless someone using these Pesukim other than at the designated time during Davening. He also presents a variant text of the above Gemara in Shabbos (שם) which indicates that the Tanna in question was not discussing Birchas Kohanim at all.
The Bach, in his commentary to the Tur (או"ח שם בד"ה גרסינן), presents a different approach, saying that the only time a non-Kohein is forbidden to recite the Birchas Kohanim is if he does so with נשיאת כפים, raised and outstretched hands, as the Kohanim do. But there is nothing wrong with a non-Kohein reciting the words of the Birchas Kohanim (even from the דוכן, as the aforementioned Tanna was willing to do) if he does not raise his hands. The Kaf HaChaim (שם ס"ק ח') suggests that the only time there is any prohibition for a non-Kohein is if his intent is to fulfill the Mitzvah of blessing the people like the Kohanim, but if his intent is simply to greet or bless a friend this way, there is no problem. The Mishnah Berurah (שם ס"ק ג') also discusses this, and concludes likewise that the prohibition is only if the non-Kohein's intent is to fulfill the Mitzvah of Birchas Kohanim, and he adds in the Biur Halacha (שם בד"ה דזר) that if one recites the Pesukim of Birchas Kohanim outside the context of Tefillah, it is certainly permissible because he is demonstrating clearly that his intent is not to fulfill the Mitzvah of the Kohanim.
It is thus permissible today according to all these authorities for parents to bless their children, or rabbis to bless their students, or friends to bless each other, even if they are not Kohanim, with the Pesukim of Birchas Kohanim. The Torah Temimah cited above quotes that the Vilna Gaon blessed people with these Pesukim, but he placed only one hand on the recipient's head, explaining that only the Kohanim in the Beis HaMikdash blessed with two hands. Rav Ovadyah Yosef (שו"ת יחוה דעת חלק ה' סימן י"ד), however, quotes many Poskim who say that one need not be concerned with this and may bless with two hands on the recipient's head.