Your question is asked by a few Mefarshim:

(1) The Tiferes Yisrael #53 answers on the basis of the Gemara Gitin 70b that if someone was "Meguyad" (Rashi explains that he was cut up) or "Tzoluv" (nailed to the stake) he can still hint that a Get should be given to his wife. One sees from this that a person in such a state is still considered alive according to the Halachah, so a Nazir would be allowed to touch him.

The Tiferes Yisrael cites the Gemara above 4b that Shimshon rendered his victims "Gosesim." A Goses is near death but still alive, so a Nazir is allowed to touch him. Similarly one could say that Shmuel only made Agag a Goses. See Tosfos 4b DH Dilma that the Gemara there does not really mean a Goses because another Gemara states (above 43a) that a Nazir must not even touch a Goses, but rather Shimshon touched them when they were almost Gosesim. (The Rosh 4b also writes that one can say that a Nazir is allowed to touch a Goses who became such through human injury, while he is not allowed to touch a Goses who became so through illness - the Gemara Sanhedrin 78a makes a similar distinction).

(2) The Mesivta edition cites other answers. Rabbi Yechezkel Landau (the author of the Noda b'Yehudah), in his Sefer Doresh l'Tzion Derashah 5 answers that by killing Agag, Shmuel fulfilled the positive commandment (Devarim 25:19) of wiping out the memory of Amalek. Therefore even though there is a negative prohibition against a Nazir touching a corpse, nevertheless the positive Mitzvah defers the prohibition ("Aseh Docheh Lo Sa'aseh").

[See Doresh l'Tzion who discusses this at length].

(3) The Pardes Yosef, on Parshas Naso, cites Rabbi Meir Shapiro (the founder of the Dafyomi cycle) who answers according to what Tosfos Yom Tov here suggests, that possibly the Mishnah learns that Shmuel was a Nazir from the fact that Shimshon was a Nazir, as stated in the Mishnah above 4a. Since the Gemara (above end 4b) states that Shimshon touched corpses therefore Shmuel also touched the dead. [However this is not the conclusion of the Tosfos Yom Tov, but rather Shimshon was allowed to touch the dead but Shmuel was not allowed to - see Rambam Hilchos Nezirus 3:13 and 3:16].

Dovid Bloom

Here is a different answer to your question. Shmuel did not actually execute Agag personally but rather he appointed a Shaliach - who was neither a Nazir or a Cohen - to do this.

Even though Shmuel 1:15:33 states "And Shmuel sliced" this does not necessary mean that he did it personally. We find something similar in Shmos 38:22 that even though the Torah states that Bezalel made the Mishkan nevertheless the Ramban there explains that this does not mean he personally made it all, but rather that he supervised its construction.

KOL TUV

Dovid Bloom