Here are some verses from start to end in Samuel on Amalek
Exodus 17:14 And the LORD said unto Moses, Write this or a memorial in a book, and rehearse it in the ears of Joshua: for I will utterly put out the remembrance of Amalek from under heaven.
Exodus 17:16 For he said,Because the LORD hath sworn that the LORD will have war with Amalek from generation to generation.
It shows the reason here
Duet 25:17 Remember what Amalek did unto thee by the way, when ye were come forth out of Egypt;
Duet 25:18 How he met thee by the way, and smote the hindmost of thee, even all that were feeble behind thee, when thou wast faint and weary; and he feared not God.
So Amalek showed no mercy
Duet 25:19 Therefore it shall be, when the LORD thy God hath given thee rest from all thine enemies round about, in the land which the LORD thy God giveth thee for an inheritance to possess it, that thou shalt blot out the remembrance of Amalek from under heaven; thou shalt not forget it.
1Sam 15:2 Thus saith the LORD of hosts, I remember that which Amalek did to Israel,how he laid wait for him in the way, when he came up from Egypt.
Therefore no mercy is being shown back
2Sam 15:3 Now go and smite Amalek, and utterly destroy all that they have, and spare them not; but slay both man and woman, infant and suckling, ox and sheep, camel and ass.
He was even angry at Saul when His wrath wasnt executed here...
1Sam 28:18 Because thou obeyedst not the voice of the LORD, nor executedst his fierce wrath upon Amalek, therefore hath the LORD done this thing unto thee this day.
Whereas Jeus said, ye have heard hate your enemies, but now I say unto love your enemies. There they were to execute his wrath, whereas we execute His mercy (avenging not ourselves but leaving room for wrath). But it also says judgment without mercy will be shown to those who have shown no mercy, And that seemed to be the start of the whole thing with Amalek (Because Amalek showed no mercy in Duet 25:17-18). It does speak of God being willing to show His wrath. And also what Saul did not do (in the very execution of it) which Samuel finishes as he says...
1Sam 15:3 And Samuel said, As thy sword hath made women childless, so shall thy mother be childless among women. And Samuel hewed Agag in pieces before the LORD in Gilgal.
Even as it says...
Lev 24:20 Breach for breach, eye for eye, tooth for tooth: as he hath caused a blemish in a man, so shall it be done to him again.
And again...
Duet 19:21 And thine eye shall not pity; but life shall go for life, eye for eye, tooth for tooth, hand for hand, foot for foot.
Then Jesus says
Mat 5:38 Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth:
Mat 5:39 But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.
His serverity and his goodness are shown, even as His wrath and his mercy are shown. In Amalek showing no mercy seemed to receive judgment without mercy. All judgment being given to the Son, being our King and captain, reminds them of what ye "have heard" but now of what He says to us.