Balaam mentions the Amalekites when he was supposed to be cursing the Israelites and says that they were “first among the nations.” (See Numbers 24:7,20) See Lord my/your Lord.
They are mentioned several times in the Book of Judges but the next big dealing is when King Saul is trying to wipe them out in order to fulfill the command of God against them. See Amalekites: Part 1 This really is the start of his downfall when he was caught up in the greed of the Amalek animals, we can only speculate why he kept Agag the king alive; possibly to ransom him for more money. So it is fitting that after the Philistines wounded him that an Amalekite is the one to kill him (2 Samuel 1:1 – 16).
David did better when it came to the Amalekites, he killed the young Amalekite who killed Saul and also wiped out the raiding party that burned his town of Ziglag. He had other battles with them and caused a great deal of destruction on them and the Edomites in general. All of this may have caused them to flee the area and find a home in Persia and Babylon. Which is where Esther and Mordecai find a man with a hatred of the Jews named Agag (See Numbers 24:7).