What is the connection between Chanukah and Miketz — as, in most years, Chanukah falls out in this Parsha?
A link between Chanukah and Parshat Mikeitz can be found in Pharoh’s dreams. In his first dream, Pharoh saw 7 bad looking cows eating 7 good looking cows. In the second dream he saw 7 good ears of grain getting swallowed up by seven thin ears of grain. Similarly, in the days of Matisyahu the Chashmonaim and his sons, Bnai Yisrael although a weaker nation, were able to overcome the much stronger nation of Greece, because of Hashem’s intervention.
יעקב אש ויוסף להבה
Ya'akov is fire and Yosef is flame.
I don't know if "flame" is the best translation, but it's the part of the fire that is not necessarily attached to the source/fuel and therefore has the potential to travel, thereby spreading the very material of the original source to everything it touches. The "fire", on the other hand, lacks this potential and is inextricably connected to the source/fuel.
Ya'akov is the source of the s'gula that spreads to the entire nation that emanates from him, but does not himself ideally leave the holy ground (physically and otherwise) to which that s'gula is bound. Yosef, on the other hand, thrives in foreign domains, in which he is able to implement the k'dusha drawn from Ya'akov Avinu and spread it to other people and places.
I cannot locate the נועם אלימלך where I saw this 6-7 years ago.
I just saw a very nice connection in "The Weak in the Hands of the Strong," by Rebbetzin Yocheved Krimsky of the Young Israel of Stamford, in the "Collected Insights" (PDF) section of this year's YU Chanuka To-Go.
It's a page-long devar Torah, worth reading in its entirety, but in essence: Par'o woke up from a dream in which weak cows swallowed up strong cows, stricken by the incongruity, and this gave both Yosef and Par'o an opportunity to publicly acknowledge God's rulership of the world. Similarly, on Chanuka, we celebrate the incongruous conquest of the strong Hellenists by the weak Jews, demonstrating again that God's really in charge.
The Vilna Gaon mentions that there is a a Remez about Chanuka in Parshas Mikeitz. At the end of Parsha, the Masora gives the number of Pasukim, but strangely for Mikeitz it also gives the number of words - 2025.
The Vilna Gaon says that it is to teach us if Poti Phera is one word or two. Secondly, 2025 is a Remez to Ner Chanukah. The gemara in Shabbos (21b) say we should light Ner Ish U'Baiso for 8 days. The Gematria of the word Ner is 250 If you multiply that by 8 days, 8 x 250 = 2000. The remaining 25 is for the 25th of Kislev.
See http://www.e-geress.org/2012/12/13/parshas-mikeitz/ for an answer to this question.