For every high priest taken from among men is appointed in matters pertaining to God for the people, to offer both gifts and sacrifices for sins. He is able to deal gently with those who are ignorant and are going astray, since he is also clothed with weakness. Because of this, he must make an offering for his own sins as well as for the people. No one takes this honor on himself; instead, a person is called by God, just as Aaron was.
Hebrews 5:1-4

In the same way, Christ did not exalt himself to become a high priest, but God who said to him,

You are my Son;
today I have become your Father,

also says in another place,

You are a priest forever
according to the order of Melchizedek.

During his earthly life, he offered prayers and appeals with loud cries and tears to the one who was able to save him from death, and he was heard because of his reverence. Although he was the Son, he learned obedience from what he suffered. After he was perfected, he became the source of eternal salvation for all who obey him, and he was declared by God a high priest according to the order of Melchizedek.
Hebrews 5:5-6

For this Melchizedek, king of Salem, priest of God Most High, met Abraham and blessed him as he returned from defeating the kings, and Abraham gave him a tenth of everything.
Hebrews 7:1-2

First, his name means king of righteousness, then also, king of Salem, meaning king of peace. Without father, mother, or genealogy, having neither beginning of days nor end of life, but resembling the Son of God, he remains a priest forever. Now consider how great this man was: even Abraham the patriarch gave a tenth of the plunder to him. The sons of Levi who receive the priestly office have a command according to the law to collect a tenth from the people—that is, from their brothers and sisters—though they have also descended from Abraham. But one without this lineage collected a tenth from Abraham and blessed the one who had the promises. Without a doubt, the inferior is blessed by the superior.
Hebrews 2:2-7

Now if perfection came through the Levitical priesthood (for on the basis of it the people received the law), what further need was there for another priest to appear, said to be according to the order of Melchizedek and not according to the order of Aaron? For when there is a change of the priesthood, there must be a change of law as well. For the one these things are spoken about belonged to a different tribe. No one from it has served at the altar. Now it is evident that our Lord came from Judah, and Moses said nothing about that tribe concerning priests. And this becomes clearer if another priest like Melchizedek appears, who did not become a priest based on a legal regulation about physical descent but based on the power of an indestructible life. For it has been testified:

You are a priest forever
according to the order of Melchizedek.

So the previous command is annulled because it was weak and unprofitable (for the law perfected nothing), but a better hope is introduced, through which we draw near to God.
Hebrews 7:11-19

Now many have become Levitical priests, since they are prevented by death from remaining in office. But because he remains forever, he holds his priesthood permanently. Therefore, he is able to save completely those who come to God through him, since he always lives to intercede for them. For this is the kind of high priest we need: holy, innocent, undefiled, separated from sinners, and exalted above the heavens. He doesn’t need to offer sacrifices every day, as high priests do—first for their own sins, then for those of the people. He did this once for all time when he offered himself. For the law appoints as high priests men who are weak, but the promise of the oath, which came after the law, appoints a Son, who has been perfected forever.
Hebrews 7:23-28

Therefore, since we have a great high priest who has passed through the heavens—Jesus the Son of God—let us hold fast to our confession. For we do not have a high priest who is unable to sympathize with our weaknesses, but one who has been tempted in every way as we are, yet without sin. Therefore, let us approach the throne of grace with boldness, so that we may receive mercy and find grace to help us in time of need.
Hebrews 4:14-16

Therefore, rid yourselves of all malice, all deceit, hypocrisy, envy, and all slander. Like newborn infants, desire the pure milk of the word, so that you may grow up into your salvation, if you have tasted that the Lord is good. As you come to him, a living stone—rejected by people but chosen and honored by God— you yourselves, as living stones, a spiritual house, are being built to be a holy priesthood to offer spiritual sacrifices acceptable to God through Jesus Christ.
1 Peter 2:1-5