Chanukah is about many things, but central to the holiday is how Hashem preserved the Jewish people, both physically and spiritually.
Antiochus, returning from Roman humiliation in Egypt and hearing that the Jews in Jerusalem had rebelled against the high priest Menelaus (who pushed for assimilation and collaborated with Antiochus), ordered his troops to crack down on the Jews. They massacred thousands. Not content with the violence, he instituted anti-Jewish laws that prohibited Torah study, brit milah, observing kashrut, and that also required avodah zarah. If he had succeeded, the Jewish people would have assimilated into the Greek world and lost their place as am segulah. For all intents and purposes we would stop being Jews and the plan and destiny of our people would have disappeared.
What is our destiny and what is Hashem's plan for us? The groundwork for an answer is found in Bereishit, in the story of Adam and Chava. In Yevamot 61a we learn that the reference to adam,"Man" is actually to the Jewish people. Carrying that further, consider the saying of Chazal, "Ma'aseh avot siman l'banim" (The deeds of the forefathers is a sign of what will happen with the children). So, what transpired with Adam and Chava is a sign for us.
What do we see in Bereishit? In the garden, we find Adam and Chava in a struggle with a wily serpent, who wants them to disregard an explicit commandment of Hashem—"of the tree of the knowledge of good and evil, thou shalt not eat of it" (2:17). They eat and Hashem pronounces a curse on all involved. In the midst of these pesukim we find Hashem speaking to the serpent, in which Hashem tells the serpent that he will be cursed, crawl on his belly, and eat dust. It is important to note that the expression to "eat dust" is elsewhere used in the Tanakh to carry the meaning of "total defeat" (cf. Micha 7:17). This is so strong that when the prophet Yeshayahu describes the days of Moshiach he says "the wolf and the lamb shall feed together, and the lion shall eat straw like the ox; and dust shall [still] be the serpent's food" (65:25). In the end, the serpent's seed is defeated and Chava's seed prevails. In describing the days of Moshiach Yeshayahu refers back to the events of creation.
And isn't this what we see in Bereishit? We read that Hashem says as much to the serpent:
"And I will put enmity between thee and the woman, and between thy seed and her seed;
he shall bruise thy head, and thou shalt bruise his heel." (3:15)
Who is the seed of the woman and who is the seed of the serpent? The serpent is often seen as the yetzer hara, but some (cf. Sforno, commentary to 3:1) see the serpent as a manifestation of haSatan. As for Chava's seed, Bereishit Rabbah (quoting Tanchuma) tells us that when Chava had a son after Hevel's death and said, "G-d hath appointed me another seed instead of Hevel" she actually looked forward to that seed which would arise from another source…King Moshiach" (Bereishit Rabbah 23). So Chava's seed is an early remez of Moshiach. As representatives, Chava and the serpent embody the fate of their seed. In this posuk, it is Chava's future seed that will strike the head of the serpent. The goal of the final crushing blow is not the seed of the serpent, but rather the serpent itself – his head will be crushed by Chava's seed. Hashem is informing the serpent that there will be constant conflict between himself and the descendants of Chava and that ultimately one of her descendants will destroy him (crush his head).
And so begins the great conflict of the ages. From the very beginning we see the conflict. The rest of Bereishit carefully traces the seed of Chava to Shet to Noach to Shem to Avraham to Yitzchak to Yaakov to Yehudah, and so on. Hashem tells us that the blessing we lost in the Garden of Eden will be restored through the seed of Chava, and that her seed (זרע ) is now one from the nation of Israel. It will be from Israel that the one who crushes the serpent's head will be born.
So what does Chanukah have to do with Moshiach?
If we understand that the preservation of the Jewish people is part of Hashem's plan to restore blessing to the Jewish people and the rest of humanity, then every attempt to destroy our people must be seen as the serpent's continuation of the conflict that began in Bereishit. Every attempt to stop God's plan – every pharaoh, every Antiochus, every Haman, every Herod that tries to kill or destroy us is serving the serpent, and every time the Jewish people have been saved and preserved it is Hashem working to keep us alive to bring the blessing to the world.
Again, what does Chanukah have to do with Moshiach?
As was stated earlier, Chanukah is about many things, but the central idea of Chanukah is about how Hashem preserved the Jewish people both physically and spiritually. The Moshiach must come from the seed of the woman. If the seed were destroyed there would be no Moshiach. Therefore, if there was no Chanukah there could not be a Jewish people and without a Jewish people there could be no Moshiach.
I am one who says that the Moshiach has come and his name is Yeshua (ישוע ). The B'rit Chadasha tells of his life and how he is the seed we have waited for, who brings victory over the seed of the serpent. He crushed the head of the serpent, though in the process he was wounded (his "heel bruised" cf. Bereishit 3:15 and Yeshayahu 52:13-53:12). If you want to read it for yourself, you may read it in Hebrew here. It is available in Yiddish here, and in English here.