(If you don't want any derush skip 1-4 but don't skip #5, the Even ha'Azel's beautiful chiddush based on a diyuk in Rambam.)
1. In Gertrude Himmelfarb's The Demoralization of Society (p 39) she quotes Hipployte Taine as saying, "The aim of every society must be a state of affiars in which every man is his own constable, until at least none other is required." In other words, "Shoftim v'shotrim titen LECHA," you have to make yourself into the shofet and shoteir. You have to develop a moral compass and police yourself.
2. Ksav Sofer notes that the parsha that speaks of the appointment of a king the Torah uses the singular voice: "Asima alay melech.... Som tasim alecha melech..." There is a din (O.C. 53:19) that when a congregation appoints a new chazzan, any single individual may object to the appointment (provided they can offer a legitimate reason for doing so). There has to be no dissension with respect to the final choice. So too, suggests the Ksav Sofer, when it comes to the appointment of a king, the people have to speak with unanimity, with one voice. There can be no objections to the selection.
I wonder if this reflects the reality. David haMelech was chased across the countryside by Shaul -- it doesn't seem that even he had the unanimous consent of the people, at least at the time of his appointment.
3. Speaking of Shaul haMelech (that last paragraph was an excuse to make a transition : )... The parsha tells us that we cannot use sorcery or fortune telling like the other nations do. Hashem instead gives us nevi'im to reveal to us what we need to know of the future. The Netziv says a chiddush: if there is no navi to consult, if there is no other recourse, and we absolutely need information, then those other means are at our disposal as well. If it's pikuach nefesh and the only way out is through sorcery, then you have to use it. Why then, asks the Netziv, was Shaul punished for consulting the witch of Endor? He was not getting an answer from the u'rim v'tumim or any other way and had no other choice?
Netziv answers that Shaul was punished because he created the situation in which he found himself. He killed the kohanim of Nov, he caused G-d not to be responsive to him, and so he was responsible for the outcome. You can't put yourself in hot water and then cry ones and expect to be excused. (There is a similar idea the Brisker Rav has on parshas Braishis -- see this post from 11 years ago.)
4. Not every king has access to a navi, and we for sure don't have access to a navi, but Chasam Sofer says the parsha has a solution for us without our having to go to fortune tellers. "V'haya k'shivta al kisei mamlachto," when the king is sitting on his throne, the Torah tells us that he has to write a sefer Torah. Chasam Sofer explains that when Klal Yisrael is on the level we are supposed to be on, our king is not sitting just on his throne -- he is sitting on Hashem's throne, as Hashem is the true king. The melech is just his top representative down here. So the parsha is not speaking about that ideal time. The parsha is speaking about the b'dieved state, when the king is just on HIS throne. There is no navi, there is no ruach hakodesh when we are in that state. So where are we supposed to get answers from? This Torah says when "k'shivto al kisei MAMLACHTO," (as opposed to malchus Hashem,) then write a sefer Torah, "V'kara bo kol y'mei chayav," and read about life in it. You want answers -- learn Torah.
A solution that applies to us as much as a king.
5. The Rambam (Melachim 3:5) writes that a king is not permitted to drink like a drunkard, but rather he is supposed to learn Torah and deal with the needs of Klal Yisrael day and night. The Rambam quotes as proof this pasuk of "v'kara bo kol y'mei chayav."
Don't all of us have to (ideally) learn Torah day and night, to the extent possible? The Rambam in hil talmud Torah ch 1 paskens this way with respect to any Jew. So why do we need a special din by a melech that he has to learn day and night?
R' Isser Zalman Meltzer in the Even ha'Azel answers that there is a difference. If you or I want to relax, we are free to sit down and have a beer, read a book, take a jog. If that leads to some bitul Torah, we are excused. Enjoying life is not assur. Bitul Torah means deliberately not learning when one has nothing else to do and no other interest at the moment. The melech, however, is different. The melech is not allowed to sit back and relax with a beer or go for a jog. He has an affirmative obligation to be engrossed in Torah and the needs of Klal Yisrael every moment, irrespective of his personal interests.
When I saw this Even ha'Azel I understood in a completely different light the statement of "man malchai? -- Rabbanan." The true kings are the Rabbis, talmidei chachamim, because only they, like kings, are engaged every moment in the dvar Hashem to the exclusion of their own interests and pleasures.