Hanukka and Yehuda
It's not very often that Hanukka falls out on Parshat Veyeishev. It's almost always on Miketz alone (and even more rare is when it falls out on Vayigash).
A few websites mention the connection between Hanukka and Yosef.
However, you can also find a connection between Hanukka and Yehuda in this week's Perasha.
In this week's Perasha, we find Yehuda demoted from his position of power וירד יהודה מאת אחיו - and Rashi, quoting Midrash, says that his brothers demoted him since it was his suggestion that caused Yosef to be sold, and it caused great pain to their father, Yaakov. So Yehuda lost his position as the king of the brothers temporarily.
Similarly, we find that the Hashmona'im took over the leadership position of the Jewish people during the time of the Hanukka story. The Hashmona'im were Kohanim - not from the tribe of Yehuda, who should not have kept the power that they ascended to. It was for this reason that they were punished by their family being wiped out.
However, everything is for the best - through the Hashmona'im, the Beit Hamikdash was rededicated and the the Jewish people had self-rule for many years.
Similarly, Yehuda's descension was for the best. As the Midrash says, "והקב"ה היה עוסק בורא אורו של מלך המשיח ... קודם שלא נולד משעבד הראשון נולד גואל האחרון" Hashem was busy creating the first light of Mashiah (via Yehuda's story) ... before the first oppressor was born, the final redeemer was born (meaning Peretz, the son of Yehuda, was the first light of Mashiah, and he was born before the Par'oh of Sefer Shemot because Hashem creates the cure before the plague).