I apologize in advance if this comes across as provocative. That is not at all my intention; I ask simply to learn.
"Kill the infidels" is a phrase often used to describe Islam's stance on disbelievers - chop their heads off, poke their eyes out, and other gruesome forms of death.
Although Judaism does not invite such severe forms of killing, they seem to believe in the similar principle of join us or die.
Rashi to Shemos 17:16 says that "as long as Amalek's memory endures, Hashem's Name and His throne are not complete." Seemingly what he is referring to is the oft-quoted statement that Amalek stands against everything Judaism is for - namely, expressing Hashem's Presence in the world. As such, we cannot exist side by side. It's for this reason that R' Aharon Kotler (Mishnas Rav Aharon, Vol. 4 pg. 137-8, explaining Sanhedrin 20b) says that we had to wipe out Amalek before building the Beis HaMikdash - as the Beis HaMikdash brings the Shechinah to the world and Amalek strives to remove it from the world, we had to destroy Amalek in order for the Beis HaMikdash to be built.
Isn't this a little extreme? In general, we have an obligation to extend peace to the nations we attack (Devarim 20:10ff), yet nowhere do we find that this applies to Amalek.1 Why is this so? Because Amalek refuses to acknowledge Hashem's sovereignty, that gives us a right to kill them? Let them get their due when Mashiach comes and the nations are punished for their sins (see AZ 2a-3b). Why can we not keep our Beis HaMikdash to ourselves and let them keep to themselves?
Had the sources said they attacked first, therefore we could attack back, that would be one thing; we find a similar discussion by Midian (Bamidbar 26,31). But that's not why we're allowed to kill Amalek, according to the above; it's because they're anti-HaShem. Why is that license to kill?
1 I am aware that Rashi to Devarim 20:10, citing Sifrei there, explains this passage as referring to optional wars, of which Amalek clearly is not, and so this point would not apply. However, the Ramban to the above passuk, as well as Rambam (Malachim 6:1,4) hold that it applies to obligatory wars as well, and thus at least according to them this point would stand. When one is supposed to call out for peace is not for this post; those curious may refer to Sefer HaChinuch 527, Tosfos to Gittin 46a, and Yerushalmi Sheviis 1:6. Also see the above Ramban inside for how he defends himself against the Sifrei.