“whose root was against Amalek.” The Hebrew can be translated that way, and it is true that the root of Ephraim was against Amalek. Joshua was from the tribe of Ephriam and as early as the wilderness wanderings with Moses, Joshua led the battles against the Amalakites (Exod. 17:8-13). The idea that Ephraim’s root was “in Amelek” is “strange” (T. Butler, Word Biblical Commentary), and forces people to say that Amalek must have controlled some of the territory inherited by Ephraim, but that is unlikely and without any proof. But even if it were the case, that would not make Ephraim’s root Amalek.
Since the Amalakites were not in this battle, one might ask why they even come up in Deborah’s poem. Apparently they come up because they were the quintessential enemies of Israel. They were the first ones to attack Israel after Israel left Egypt (Exod. 17:8) and God said He would make war on them continually (Exod. 17:16). So here in Judges 5, although the Canaanites were the ones who oppressed Israel for 20 years (Judg. 4:3), Deborah mentions them in her poem for effect and emphasis.
“After you, Benjamin, among your peoples.” This is not a complete sentence in English, and it is not a complete sentence in Hebrew. In fact, the versions are divided as to what it means and how to translate it. The two primary interpretations are that it is saying that Benjamin followed Ephraim into battle ( CJB; HCSB; NIV, NJB), and that Ephraim followed Benjamin into battle (ESV; NASB; NET; NLT). Judges 5 is Hebrew poetry, and like most poetry, some sentences are incomplete and vocabulary words are used in unusual ways, making this chapter difficult to understand in a precise way. The text is simply unclear about who followed who into battle.
“the officer’s staff.” The Hebrew text is unclear because the phrase is used only here, and so the English versions translate it in many different ways, including “the pen of the writer” (KJV); “census-counter’s staff “ (Fox); and “staff of office” (NASB). Zebulun was in the heat of the battle (Judg. 5:18), so they apparently were taking some form of leadership or forward role in the fighting, so “officer’s staff” seemed logical and was similar to many other English versions.