Gemara

(Rav Acha bar Bizna): There was a harp above David'd bed. At midnight, the north wind would make it play. David would wake up and learn Torah until morning - "Urah Chevodi Urah ha'Nevel v'Chinor A'irah ha'Shachar."

Berachos 3a (R. Yitzchak bar Shmuel): There are three Mishmaros (divisions of the night). In each Mishmar, Hash-m roars like a lion "woe to My children, for due to their sins I destroyed My house, burned My sanctuary, and exiled them among the nations of the world!"

(Beraisa - R. Eliezer): There is a sign for each Mishmar. A donkey brays in the first, dogs bark in the second, and babies nurse in the third.

The signs are for the end of the first Mishmar, the middle of the second, and the beginning of the last Mishmar.

Yevamos 72a (Beraisa): All 40 years in the Midbar, each night the north wind blew at midnight - "at midnight, Hash-m killed every firstborn."

Question: How do we learn from this verse?

Answer: It shows that Hash-m is appeased at midnight.

Rishonim

Rosh (Berachos 1:2): It is proper for everyone who fears Shamayim to be pained at the times the Mishmaros change, and supplicate over the Churban - "Kumi Roni va'Laylah l'Rosh Ashmuros."

Poskim

Shulchan Aruch (OC 1:1): One should invigorate himself like a lion each morning to rise to serve Hash-m. He should arouse (rise before) dawn.

Taz (2): The Tur says (that David said) 'I awaken the dawn. The dawn does not awaken me.' If one sanctifies himself, Hash-m sanctifies him. David praised himself that he aroused himself and did not need arousal from above. For people like us, the most we can hope for is the first part, to arouse oneself through rising before dawn, and then receive arousal from above.

Mishbetzos Zahav: One who is sleeping cannot invigorate himself! Rather, one should eat light foods so he will not sleep much.

Gra (3): The Yerushalmi says that David rose before dawn because his honor was insignificant in front of Hash-m's honor.

Prishah OC 1:10 The Yetzer ha'Ra gets stronger before dawn, when one must rise to prepare himself to serve Hash-m.

Magen Avraham (1, citing the Shlah): It is a great attribute to join day and night through Torah or Tefilah, both at dawn and in the evening

Kaf ha'Chayim (24): One hour of learning before dawn is worth more than several hours during the day

Mishnah Berurah (1): One was created to serve Hash-m. Even if the Yetzer ha'Ra entices in winter and says 'you cannot rise. It is too cold', or in summer it says 'you cannot rise. You did not sleep enough', he should not heed it. He should invigorate himself, and think 'if I needed to serve a mortal king, I would be so zealous to rise early to serve him. How much the more so I must do so to serve the King of kings, ha'Kadosh Baruch Hu!

Rema: In any case he should not delay after the Tzibur prays... When lying on his bed, he should wake from his sleep with Zerizus, to serve his Creator.

Gra (4): Morning sleep removes a person from the world (Avos 3:10), i.e. if he sleeps (Rashi on Avos; Avos d'R. Noson 21:1 - intends to sleep) past the time for Kri'as Shema.

R. Akiva Eiger: Even if he cannot pray with a Tzibur, he should not delay past when the Tzibur prays.

Shulchan Aruch (2): One who rises early to supplicate in front of his creator should match the times when the Mishmaros (watches of the night) change, which are after a third of the night and after two thirds of the night and at the end of the night. Prayer at these times over the Churban and exile are accepted.

Magen Avraham (4): The Rikanti (Bereishis 14:1 DH Sefer) says that some say that the time to request mercy is at the end of the night. I say that midnight is best to request mercy for Keneses Yisrael and her children, and at the end of the night he requests his needs. This must be with Nefilas Apayim (falling on one's face).

Machatzis ha'Shekel (DH uvi'Gemara): There are different opinions of how to calculate midnight. R. Eliezer gave a sign for the middle of the second Mishmar. Surely this is so people will know the ideal time to supplicate about the Churban! The Rashba says that it is so people will know the last time for eating (certain) Kodshim. He said so because the Zohar was not known in his days. R. Eliezer said that Hash-m laments the Churban at these times. The Rosh brought this, and right afterwards teaches that one should be pained and supplicate over the Churban when the Mishmaros change.

Mishnah Brurah (9): The Mekubalim greatly praise the attribute of rising at midnight, i.e. 12 hours after the middle of the daylight hours. It is best to mourn the Churban slightly before midnight, and to learn Torah after midnight, and to request one's needs at the end of the night. If it is close to sunrise, he should say the Tehilim in Tikun Chatzos and skip the Kinos. If one is weak, or if rising before dawn will cause him to sleep during Tefilah, he should sleep as much as he needs, but in any case he should rise an hour, or at the least half an hour, before the attendant calls people to go to Shul, in order to prepare himself properly.

Kaf ha'Chayim (16 and Salmas Chaim OC 1): Tikun Chatzos has precedence over Selichos if there is not time for both.

Salmas Chaim (2,3): One can be Yotzei with any amount, e.g. Al Naharos Bavel and Mizmor l'Asaf. One can be Yotzei with Slichos itself, which laments Galus of the Shechinah and Tzadikim, as long as he intends for it. If one can rise at Chatzos, he must do so!

Doleh u'Mashke ((ha'Gaon R. Chayim Kanievsky, Shlita), p. 65): It is not an absolute Chiyuv, so if one can pray k'Vatikin (begin Shemoneh Esre at sunrise), or say Tikun Chatzos (but not both), he may so as he desires.

Kaf ha'Chayim (7): The Zohar greatly praises one who is awake at exactly midnight. One should try to avoid being asleep then, lest he taste death and a Ru'ach of Tum'ah cling to him. One who sleeps during the first six hours of the night does not taste death. Also one who rises at midnight, says Berachos and Tikun Chatzos does not taste death even if he goes back to sleep.

Kaf ha'Chayim (8): Even one who did not rise at midnight, rather, an hour or two later, must rise then. He may not say 'since I did not rise at midnight, I will keep sleeping.' Rather, the earlier he rises, the better, lest he taste death longer and more Ru'ach of Tum'ah cling to him. If even this he cannot do, at least he should rise before dawn to say Tikun Chatzos and join day and night through Torah.

Kaf ha'Chayim (9): Ru'ach of Tum'ah does not apply on Shabbos, but in any case it is good to rise at midnight to fulfill his habit, and also because the latter half of the night is the best time to learn Torah.

Kaf ha'Chayim (10): It is ideal to stay up from midnight until dawn, learning Torah. If one cannot, he cries over the Churban for half an hour, then learns Torah for a Sha'ah (hour, or a while) and goes back to sleep. He should wake up again a half-hour before dawn in order to join day and night through Torah.

Kaf ha'Chayim (15): Women are not accustomed to saying Tikun Chatzos, for in their youth they are busy raising children, and once they are used to not saying it, they continue. Even many men do not say it, only those who (greatly) fear Shamayim! If a woman wants to say it, she will be blessed.

Mishnah Berurah (561:103): Bein ha'Metzarim (in the three weeks before Tish'ah b'Av), right after midday one should cry over the Churban for about a half hour.

Yalkut Yosef (1:45): We say Tikun Chatzos at midday only Bein ha'Metzarim.