Why is Parshat Zachor always read on the Shabbat prior to Purim? The Mishnah Berurah (O.C 685:1:1, citing the Gemara) states that this is because Parshat Zachor reminds us to wipe out Amalek, and Haman of the Purim story was related to Amalek. The connection, then, is that Haman, a later descendant of Amalek, was wiped out, just as we are commanded to wipe out the entire nation. This is a fine answer, but nonetheless, we may perhaps find another explanation.
Rashi (on Devarim 25:17) comments that Amalek represents “Mikreh,” coincidence (as hinted to by the word “Karecha” in that Pasuk). Amalek denies the existence of Hashem and his role in coordinating the happenings of the world, claiming that all that occurs is merely coincidental.
Rav Avrohom Gordimer explains that the major theme of Purim is the exact opposite of this – the recognition that God controls everything, even though we may not perceive His hand in what happens. Purim demonstrates how Hashem’s Hashgachah, providence, operates even in a world without open miracles. Even though the occurrences in the Purim story appear to be normal and natural, Hashem’s hand is really behind every event that transpires.
Amalek claims that things happen completely due to fate and chance. The Purim story counters this belief: in truth, everything that happens, whether Hashem’s hand in the matter can be seen or not, is all part of His well-designed plan. That, Rav Gordimer suggests, is yet another connection between Parshat Zachor and Purim.