When does the fast begin?
The fast begins at Alos Hashachar of the day of the 10th of Teves. This is approximately two fluctuating hours prior to sunrise. Thus even if one was awake the entire night, he must stop eating and drinking two fluctuating hours before sunrise.
One who woke up before Alos: If one slept [even on the couch] and then woke up prior to Alos, he may not eat unless he stipulated prior to going to sleep that he plans to eat after awaking before Alos. He may however have a drink, even if he did not stipulate before going to sleep, although initially it is proper to stipulate before going to sleep even if one only plans to drink and not to eat. If one did not sleep a set sleep but rather slightly dozed off, it is considered as if he has not slept and he may continue to eat until Alos.
May one eat a lot of food prior to a fast in order so he does not feel hungry?
The Poskim rule that one may not eat more than usual prior to a fast as this defeats the entire purpose of feeling oppression on that day. Nevertheless one who is weak natured and needs to eat more in order to have strength is to do so.
 Definition of fluctuating hours: This means that the hours fluctuate in the winter and summer. Some Poskim rule this means it fluctuates in terms of Zmaniyos, meaning that it depends on the amount of hours in the day. Thus in the summer, the hours will be longer [between 120-150 minutes for two hours] while in the winter they will be shorter [between 90-120 minutes for two hours]. [So rules Minchas Cohen 2/6; Rama 233; Peri Chadash 58] However the Alter Rebbe and Gr”a both rule that it follows not the amount of hours in the day but rather the degree of distance of the sun from the horizon. [Admur in Seder Hachnasas Shabbos; Gr”a in 261; See Shut Mahrshag 2/34 quoted in Piskeiy Teshuvos 89/2 footnote 59 that this is the way we rule.] Thus those that hold that Alos is 72 minutes it ends up being in Tishrei and Nissan 16.1 degrees from the horizon and the amount of time it takes the sun to travel to the horizon fluctuates between winter and summer. See Piskeiy Teshuvos 89/2
Other opinions: Some Poskim rule we always measure the hours as set hours and hence there will always be only 120/90/72 minutes between Alos and sunrise at all times. [Admur 89/1; Birkeiy Yosef 261/1; Peri Megadim 261 A”A 9; Derech Hachaim; Siddur Yaavetz; Machatzis Hashekel 235/3]
 Based on the ruling of Harav Hagaon Avraham Chaim Naah printed in Shiureiy Mikveh 37; Yagdil Torah Tzemach Tzedek 23 p. 23 [see there for a full organized summary on the subject]
Background of opinion of Admur according to the Gra”ch Naah-2 hours: Alos Hashachar begins 2 Zmaniyos hours prior to sunrise. The calculation is as follows: There are 5 Mil between Alos and Neitz [as rules Ula in Pesachim 93b]. Each Mil is 24 minutes [as rules Rambam in Pirush Hamishnayos Pesachim 3/2]. Thus 24 minutes per Mil x 5 Mil between Alos and Neitz equals 120 minutes. [This follows the ruling of Admur in 459/10 and Siddur and so rules regarding 24 minutes per Mil: Peri Chadash Y.D. 69/26; Kitzur SHU”A 36/11. However in 89/1 and 261/5 Admur rules that there is only 4 Mil between Alos and sunrise, hence there is only 96 minutes between Alos and sunrise. As well although in 459/10 Admur rules that the day begins from sunrise and ends by sunset in 89/1 he rules that it begins from Alos until nightfall. Nevertheless the final ruling of Admur follows the ruling of the Siddur in which Admur rules like in 459/10.]
Opinion of Shulchan Aruch and majority of Poskim-72 minutes: Alos Hashachar begins 72 minutes prior to sunrise. This is based on the calculation that there are 18 minutes per Mil [as rules Terumos Hadeshen 123; Michaber 459/2; Yoreh Deah 69/6; Rama 261/1] and there are 4 Mil between Alos and Neitz [as rules Rebbe Yehuda in Pesachim 93b] The following Poskim rule like this opinion: Rashal Pesachim 2a; Minchas Kohen 2/6; M”A 89/2; Levush 261 and 459; Admur in 89/1 and 261/5; Derech Hachaim; M”B 89 in Biur Halacha “Veim”; 58 Biur Halacha “Kemo”; and chapters: 92; 163; 235; 261; 459
Opinion of some Achronim-90 minutes: Alos Hashachar begins 90 minutes prior to sunrise. This is based on the calculation that there are 22.5 minutes per Mil [as rules Mahril in Hilchos Pesach] and there are 4 Mil between Alos and Neitz [as rules Rebbe Yehuda in Pesachim 93b] So rules the following Poskim: Gr”a 459; Chok Yaakov 459/10; Chasam Sofer in glosses 89
 M”B 564/3
 Michaber 564/1
 Rama 564/1 being that it is common to drink upon awakening in middle of the night and hence it is considered as if one stipulated even if one did not explicitly do so.
Other Opinions: The Michaber ibid rules one must stipulate even in order to drink upon awakening.
 M”B 564/6 in name of Elya Raba. However if one did not stipulate he may nevertheless drink. [M”B ibid and in Shaareiy Tziyon 564/7]
 564/1; M”B 564/2-3
 Kitzur Shlah; Nehar Shalom 550/2; Elya Raba 563/1; Bigdei Yesha 564
 Kaf Hachaim 550/11