Torah Temimah: Because Yisrael did not really need a king - since Hashem is their King, ¹ and it is only because Hashem foresaw that they would (reject Him and) ask for a king that He turned the request into a Mitzvah.

Yerushalmi (Sanhedrin 2:6): Because Hashem only granted them a king because they aasked for one. ²

Rashi (on Pasuk 14) ¹ R. Bachye #1, Targum Yonasan and Targum Onkelos: It is a Mitzvah to appoint a king (after consulting Hashem - Targum Yonasan). ²

Ramban #1 (citing the Sifri): To teach us that if the current king dies it is a Mitzvah to appoint another king in his place.

R. Bachye #2 (according to Kabalah): Hashem's Throne comprised three legs (Chesed, Gevurah and Tiferes) - represented by the three Avos - and was short of the fourth leg - until David was crowned. It was therefore imperative to have a king ³ in order to complete Hashem's Throne. ⁴

R. Bachye (citing Sotah 41b): It implies that one may only appoint a king whose father and mother are from Yisrael. ¹

Ba'al ha'Turim: "mi'Kerev Achecha" has the same Gematriya as ?mi'Sheivet Yehudah'.

Sifri: It implies that the king must be from Eretz Yisrael and not from Chutz la'Aretz. ²

Yevamos, 120a; "Alecha" implies that a Ger who is a Dayan is not permitted to judge a Yisrael but he is permitted to judge his fellow-Ger. ³

Kidushin, 76b #1: It means that his mother is a Bas Yisrael ⁴ (even though his father is a Ger). ⁵

Kidushin, 76b #2: It implies that every important appointment ⁶ should be from you - someone whose mother is a Bas-Yisrael - and not a Ger. ⁷

Bava Kama, 88a: It precludes anyone who does not have a Yichus - such as an Eved Cana'ani - from becoming king. ⁸

Yerushalmi Kidushin, 4:5: To include all important appointments in the current restriction - such as heads of police, Gaba'ei Tzedakah, the scribes of Beis-Din and those who whip the malcreants. ⁹

Sanhedrin, 22a:

Kidushin, 32b: See answer #1. From the king's perspective, it prohibits him from foregoing his Kavod - 'Melech she'Machal al Kevodo, Ein Kevodo Machul'. ¹

Ramban #1: It implies that the appointment must be made through a Navi or via the Urim ve'Tumim. ¹

Ramban #2 and R. Bachye: It implies that Malchus and all other Midos are min ha'Shamayim. ^{2 3}

Targum Yonasan: It implies that, before choosing a king, one should consult Hashem. ⁴

Ramban #2 (citing the Sifri): It implies "Melech", 've'Lo Malkah' - to disqualify a queen from sitting on the throne. ¹

R. Bachye #2 (citing Sanhedrin 20b): To obligate us to hold the king in awe ² (not to ride on his horse, sit on his throne or use his scepter [or any of his household articles -R. Bachye], and not to see him when he is having a haircut, when he is naked or when he is in the bath-house -Sanhedrin 22a).

Kidushin 32b: To teach us that a king cannot forego his honor ³ - 'she'Tehei Eimaso alecha'.

Sanhedrin, 22a and Sifri #1: To teach us to be constantly awe-stricken before the king, and that consequently, one may not ride on his horse, sit on his throne, use his scepter or see him whilst he is having a hair-cut, when he is naked or whilst he is in the bath-house. ⁴

Rashbam: It is a prohibition against appointing a Nochri king to fight their battles - because he will lead them astray to serve Avodah Zarah. ¹

Oznayim la'Torah #1: It means, not literally a Nochri, but somewhat who is estranged from the Torah. ²

Oznayim la'Torah #2: It means that, even if a Nochri offers to rule over Yisrael, they should refuse his offer. ³

Ramban #1: To teach us that, even if, for some reason, Hakadosh-Baruch-Hu declines to answer through Nevi'im, it is forbidden to crown a Nochri king.