The first is to understand the Psalm as David describing himself, and to see that poetic descriptions describing David also apply to the Messiah.
The second is to interpret the verse as not about David, but as either David or someone in David's court directly describing the Kingly Messiah, as distinct from David.
The third is to note Messianic aspects to Psalm 22, distinct from the issue of the narrator's identity.
Let's start by considering the first approach, which applies the Psalms' descriptions of David to the Messiah.
By introducing the Psalm with the heading "A Psalm of David", the chapter makes it look like it is by, belonging to, or about David.
2 Samuel 23:1 calls David the "Sweet Singer" or "Psalmist" (Zamirot) of Israel. David was described also as singing to King Saul as a youth. It's natural then to associate David the figure with the Psalms attributed to him, as if these were the kinds of songs that he would acquire and sing.
David as a youth, known to sing with harp and lyre
Imagining that David sings the words in Psalm 22, one would see him as talking about himself poetically, as when he sings "My God, My God, Why Hast Thou Forsaken Me?" At times in his life, according to the Bible, David was attacked or hunted by enemies and he had to go into hiding. Psalm 3 for example, specifically says that it was about David when David was under repression.
Further, in numerous places in the Bible, "David" is used as a name or metaphor for the Messiah.
Isaiah 55 says:
The word "peoples" used here commonly refers to the nations, especially those of the world in general.3 Incline your ear, and come unto me: hear, and your soul shall live; and I will make an everlasting covenant with you, even the sure mercies of David.
4 Behold, I have given him for a witness to the people, a leader and commander to the peoples(lə-’um-mîm).
Checking the word um-mim, we see that it refers to foreign peoples in the Psalms, as in:
Psalm 2:1 Why do the nations rage, And the peoples (ū-lə-’um-mîm) plot a vain thing?
For the other usages, click here:
http://biblehub.com/hebrew/3816.htm
David did not actually command the nations of the world in real life, so in what way does Isaiah 55 see him as a commander of the nations? The Messiah is the one whom Jewish tradition commonly sees as the divinely-appointed Jewish ruler over the world.
Another example is Ezekiel 34:
David had long been dead by the time Ezekiel wrote this, so of whom was Ezekiel writing and calling David?23 I will establish one shepherd over them, and he shall feed them—My servant David. He shall feed them and be their shepherd.
24 And I, the Lord, will be their God, and My servant David a prince among them; I, the Lord, have spoken.
Ezekiel 34's context is a future prophecy about freeing Israel from the rule of foreign "shepherds". In Jewish tradition, the Messiah is the one who would lead Israel after its freedom from foreign domination, so Ezekiel 34's reference to David is a metaphor for the Messiah.
David in real life was described as a shepherd in his youth, so it's understandable that Zechariah 11-13 is also talking about a Davidic Messiah when it talks about the Good Shepherd whom he predicts would rule Israel.
Ezekiel uses the metaphor of David again later in Chp. 37:24-25 in his prediction about David being Israel's "prince forever":
Jeremiah 30:9 predicts: "But they shall serve the Lord their God, And David their king, Whom I will raise up for them.""My servant David will be king over them, and they will all have one shepherd; and they will walk in My ordinances and keep My statutes and observe them.
"They will live on the land that I gave to Jacob My servant, in which your fathers lived; and they will live on it, they, and their sons and their sons' sons, forever; and David My servant will be their prince forever.
Hosea 3:5 says: "Afterward the children of Israel shall return and seek the Lord their God and David their king."
In what sense would the Israelites seek David their king if he were dead?
Either these passages about David are speaking of him metaphorically or talking about a post-resurrection future kingdom ruled by a resurrected David.
The future situation when David's throne would rule the world and its nations was commonly seen in Jewish tradition as a Messianic situation, so these passages would be interpreted Messianically too, with David being a metaphor for the Messiah.
Naturally then, both Christian theologians and nonChristian rabbis have a longstanding common practice of applying the Psalms to the Messiah. And since David's Psalm 22 describes the narrator as killed and resurrecting, then by extension its Davidic image would apply as a metaphor for the Davidic Messiah as well.
A case in point is the 6th-7th century rabbinic commentary Pesikta Rabbati or "Great Section" (chapters 34-37), which considers Psalm 22 to refer to the Messiah son of David suffering "for the sake of Israel." Jewish Virtual Library describes the Pesikta Rabbati as
http://www.jewishvirtuallibrary.org/jso ... 15652.htmla medieval Midrash on the festivals of the year... The word pesikta means "section," and this Midrash consists of a series of separate sections, on the pentateuchal and prophetic lessons of festivals... It is called Rabbati ("the greater") probably in contrast to the earlier *Pesikta (de-Rav Kahana).
...
So far it has been discovered that two major sources are represented in the Pesikta Rabbati: (1) the Pesikta de-Rav Kahana, and (2) the Tanḥuma-Yelammedenu.
...
The source material is all Palestinian, and though the precise date and place of compilation have not yet been fixed with certainty, modern scholarly opinion tends to view the Pesikta Rabbati as a Palestinian work of the sixth or seventh century.
Click here for the Hebrew of Pesikta Rabbati:
https://www.sefaria.org/Pesikta_Rabbati ... s&lang2=en
You should be able to click on the right hand for citations from the Tanakh used. According to the webpage, Pes. Rab. chps. 36 and 37 cites Psalms 22 v.16 and v.8, respectively.
Here is an English translation:
The main counterclaim that I have seen is that Pesikta Rabbati is not talking about the Messiah Son of David and attributing suffering to him, but about a different Messiah, the "son of Ephraim". However, if you check the underlined part above, you should see that indeed the Pesikta Rabbati refers to the Davidic Messiah, the Son of David, and explains that David was crying because of the Davidic son's suffering.[Chapter 36]
[At the time of the Messiah’s creation], the Holy One, blessed be He, will tell him in detail what will befall him: There are souls that have been put away with thee under My throne, and it is their sins which will bend thee down under a yoke of iron and make thee like a calf whose eyes grow dim with suffering, and will choke thy spirit as with a yoke; because of the sins of these souls thy tongue will cleave to the roof of thy mouth. Art thou willing to endure such things?
The Messiah will ask the Holy One, blessed be He: Will my suffering last many years?
The Holy One, blessed be He, will reply: Upon thy life and the life of My head, it is a period of seven years which I have decreed for thee. But if thy soul is sad at the prospect of thy suffering, I shall at this moment banish these sinful souls.
The Messiah will say: Master of the universe, with joy in my soul and gladness in my heart I take this suffering upon myself, provided that not one person in Israel perish; that not only those who are alive be saved in my days, but that also those who are dead, who died from the days of Adam up to the time of redemption; and that not only these be saved in my days, but also those who died as abortions; and that not only these be saved in my days, but all those whom Thou thoughtest to create but were not created. Such are the things I desire, and for these I am ready to take upon myself [whatever Thou decreest]….
2. During the seven–year period preceding the coming of the son of David, iron beams will be brought and loaded upon his neck until the Messiah’s body is bent low. Then he will cry and weep, and his voice will rise up to the very height of heaven, and he will say to God: Master of the universe, how much can my strength endure? How much can my spirit endure? How much my breath before it ceases? How much can my limbs suffer? Am I not flesh–and–blood?
It was because of the ordeal of the son of David that David wept, saying My strength is dried up like a potsherd (Ps. 22:16). During the ordeal of the son of David, the Holy One, blessed be He, will say to him: Ephraim, My true Messiah, long ago, ever since the six days of creation, thou didst take this ordeal upon thyself. At this moment, thy pain is like My pain. Ever since the day that the wicked Nebuchadnezzar came up and destroyed My House and burned My Temple and banished My children among the nations of the world—and this I swear by thy life and the life of My own head—I have not been able to bring Myself to sit upon My throne. And if thou dost not believe Me, see the night dew that has fallen upon My head, as is said My head is filled with dew, My locks with the drops of the night (Song 5:2).
At these words, the Messiah will reply: Now I am reconciled. The servant is content to be like his Master.
[Chapter 37]
It is taught, moreover, that in the month of Nisan the Patriarchs will arise and say to the Messiah: Ephraim, our true Messiah, even though we are thy forbears, thou art greater than we because thou didst suffer for the iniquities of our children, and terrible ordeals befell thee, such ordeals as did not befall earlier generations or later ones; for the sake of Israel thou didst become a laughingstock and a derision among the nations of the earth; and didst sit in darkness, in thick darkness, and thine eyes saw no light, and thy skin cleaved to thy bones, and thy body was as dry as a piece of wood; and thine eyes grew dim from fasting, and thy strength was dried up like a potsherd—all these afflictions on account of the iniquities of our children, all these because of thy desire to have our children benefit by that goodness which the Holy One, blessed be He, will bestow in abundance upon Israel. Yet it may be because of the anguish which thou didst greatly suffer on their account—for thine enemies put thee in prison—that thou art displeased with them!
He will reply: O Patriarchs, all that I have done, I have done only for your sake and for the sake of your children, for your glory and for the glory of your children, that they benefit from that goodness which the Holy One, blessed be He, will bestow in abundance upon them—upon Israel.
The Patriarchs will say to him: Ephraim, our true Messiah, be content with what thou hast done, for thou hast made content the mind of thy Maker and our minds also.
Then—so taught R. Simeon ben Pazzi—the Holy One, blessed be He, will lift the Messiah up to the heaven of heavens, and will cloak him in something of the splendor of His own glory as protection against the nations of the earth, particularly against the wicked Persians. He will be told: Ephraim, our true Messiah, be thou judge of these and do with them what thy soul desires, for the nations would long since have destroyed thee in an instant had not God’s mercies been exceedingly mighty in thy behalf, as is said Ephraim is a darling son unto Me—is he not as a child that is dandled? For as often as I speak of him I do earnestly remember him still; therefore My heart yearneth for him, in mercy I will have mercy upon him, saith the Lord (Jer. 31:20). Why does the verse speak twice of mercy: In mercy I will have mercy upon him? One mercy refers to the time when he will be shut up in prison, a time when the nations of the earth will gnash their teeth at him every day, wink their eyes at one another in derision of him, nod their heads at him in contempt, open wide their lips to guffaw, as is said All they that see me laugh me to scorn,’ they shoot out the lip, they shake the head (Ps. 22:8); My strength is dried up like a potsherd; and my tongue cleaveth to my throat; and thou layest me in the dust of death (Ps. 22:16).
Moreover, they will roar over him like lions, as is said They open wide their mouth against me, as a ravening and roaring lion. I am poured out like water, and all my bones are out of joint; my heart is become like wax; it is melted in mine inmost parts (Ps. 22:14-15). They will growl over him like lions who lust to swallow him, as is said All our enemies have opened their mouth wide against us. Terror and the pit are come upon us, desolation and destruction (Lam. 3:46-47)
(Pesikta Rabbati, William G, Braude, Translator (New Haven: Yale University, 1968), Volume II).
Later in the passage, the Pesikta has the rabbis calling the Messiah "Ephraim", but the use of another title doesn't mean that this was not the Davidic Messiah. In the traditional Jewish view of the Messianic era, Israel, the nation made of its twelve tribes (including Ephraim that was part of the Tribe of Joseph), is seen as restored and ruled by the Messiah. Ezekiel 37 gives a similar image, with the Davidic Messiah ruling the two sticks that are united.