From antiquity, Amalek appears as the people of the crushed head because of the apparent association with the serpent in the Garden of Eden, and so the imagery stems from the Book of Genesis where the head of the serpent will be crushed by the heel.
To begin, the phrase "עם מלק" ("'am malak") appears to mean people of the broken head. The 19th Century Hebraist John Lange provides the following.
9 And these are the generations of Esau the father of the Edomites, in mount Seir: 10 These are the names of Esau’s sons; Eliphaz the son of Adah the wife of Esau; Reuel the son of Bashemath the wife of Esau. 11 And the sons of Eliphaz were, Teman [right side, southlander], Omar [Gesenius: eloquent; Fürst: mountain-dweller], Zepho [watch], and 12 Gatam [Gesenius: puny, thin; Fürst: burnt, dry valley] and Kenaz [hunting]. And Timna [restraint] was concubine to Eliphaz, Esau’s son: and she bare to Eliphaz, Amalek [From עַם מָלַק , a nation of head-breakers, spoilers? Lange: laboring, licking up]: these were 13 the sons of Adah, Esau’s wife. And these are the sons of Reuel; Nahath [going down, evening], and Zerah [rising, morning], Shammah [wasting; Fürst: report, call], and Mizzah [Gesenius: fear; Fürst: perhaps joy, rejoicing]: these were the sons of Bashemath, Esau’s wife.
According to the 18th Century Hebraist, Wilhelm Gensenius, the better translation of "עם מלק" ("'am malak") may be, people of the crushed head.
מָלַק TO BREAK, TO CRUSH (einknicken), Levit. 1:15; 5:8. (Cognate is the root פָּרַק, whence this has arisen, the letter r being softened into l, p, and m interchanged. Syr. and Chald. מלג vellicavit.) LXX. ἀποκνίζω, to cut off with the nail (abkneipen), contrary to the express words of the Hebrew text, יַבְדִּיל וְלֹא Lev. 5:8.
During the First Century of the Current Era, the Hellenistic Jewish philosopher, Philo of Alexandria suggested the correlation between Amalek and the serpent as follows.
. . . that is to say, Amalek; which name, being interpreted, means, the people licking. For he does, of a verity, devour the whole soul, and licks it up, leaving no seed behind, nor anything which can excite virtue.
Philo wrote in the First Century, and therefore his source of information appears to stem from interpretations relating to the serpent licking the dust. For example, the serpent licking the dust appears among the prophets of the Hebrew Bible:
Micah 7:17 (NASB)
17 They will lick the dust like a serpent,
Like reptiles of the earth.
They will come trembling out of their fortresses;
To the Lord our God they will come in dread
And they will be afraid before You.
The Amalekites therefore are the people of the crushed head of the serpent as the narrative of Genesis makes this apparent.
For example, when the Israelites escaped Egypt and entered the Promised Land, the Amalekites had “tripped them up.” That is, they attacked the Israelites at their weak spot, or at their hindermost part or “tail,” which was comprised of those who had lagged behind (Deut 25:17). The Hebrew word for the hindermost part of the body is עָקֵב, which is used in the Hebrew Bible to refer to the rear echelon of an army (Genesis 49:19 and Joshua 8:13). That is, the rear echelon of an army is their hindermost part. The Amalekites attacked Israel at their hindermost part. The exact same Hebrew word is also found in Genesis 3:15, where the serpent bites the “hindermost part” of the seed of the woman -- that is, the serpent attacks his heel, since the heel is physiologically the hindermost part of the human body. The moral and spiritual seed of the serpent is therefore the enemy of God. “Amalek” therefore is the enemy of the seed of God (Israel), who “trips up” at the most vulnerable time and opportunity.
In this regard, the Amalekites within the land of Canaan were not the seed of the Amorites, the Hittites, the Perizzites, the Canaanites, the Hivites or the Jebusites, but of Esau, who was the twin brother of Jacob (Genesis 36:12). When Esau was born, his twin brother tried “to trip him up” at birth, and therefore the twin brother was named “Jacob.” Jacob “the heel” was born from the seed of promise through Abraham. “Amalek” therefore is not biological, but moral and spiritual (since Esau and Jacob were born from the same biological parents). Please note that “Jacob” (יַעֲקֹב) and “heel” (עָקֵב) are therefore related etymologically. That is, the plain and normal meaning of the word יַעֲקֹב “Jacob” is qal imperfect of עָקַב, which means he will heel (and therefore crush).
In summary, the head of the spiritual seed of Amalek (serpent) is crushed by the head of the spiritual seed of the Jacob-heel (Israel). Thus the Amalekites are the people of the crushed head of the serpent.
Gesenius, Wilhelm, & Tregelles, Samuel P. (2003). Gesenius’ Hebrew and Chaldee Lexicon to the Old Testament Scriptures. Grand Rapids: Baker Book House: 479.
Lange, John P., Schaff, Philip, Lewis, T., & Gosman, A. (2008). A Commentary on the Holy Scriptures: Genesis. Grand Rapids: Zondervan, 572.
Yonge, Charles D. with Philo of Alexandria. (1995). The Works of Philo: Complete and Unabridged. Peabody: Hendrickson, 71.