A couple of years ago, a group of Jewish scholars published the second edition of The Jewish Study Bible (New York, New York: Oxford University Press, 2014). It's worth noting how much Christianity is corroborated by an Old Testament translation and commentary that's so liberal and non-Christian. Take their treatment of Psalm 22, for example.
We're told that the psalm is about "a person in dire straits, apparently a serious illness" (1290). There are references to Jewish traditions that see the passage as being about "a lament by David over the future exile" or "the threat against the Jews by Haman in the book of Esth." (ibid.) All of those explanations of the passage are much weaker than the traditional Christian view.
The psalm is attributed to David, but there's nothing in David's life that even comes close to aligning with the passage. It aligns far better with Jesus' crucifixion than with anything we know of from David's life. Since the Messiah was expected to be a descendant of David, the passage makes the most sense as a Messianic prophecy written by David. It's highly unlikely that there's some naturalistic explanation for an unknown ancient Jew, who wouldn't have had much interest in crucifixion, writing something so much resembling a Roman crucifixion scene, then having his psalm attributed to David, who was so closely associated with the expected Messiah and who never experienced anything close to what the psalm describes.
If the psalm is about an individual who's sick, why would many enemies (verse 12) gather around his sickbed and want the clothing of somebody who was ill? The psalm asks God for deliverance from enemies (verses 20-1), but never from illness. The reference to casting lots for clothing makes more sense in the context of a crucifixion, in which a person's clothing would be his only possession at hand and would have been taken from him just before his being crucified unclothed. The passage makes much more sense as a public execution scene, like a Roman crucifixion, which could easily involve the gathering of a large number of enemies (which would be less likely in the context of a sickbed in somebody's home), the casting of lots for clothing, the stretching out of limbs, the piercing of hands and feet, etc.
The Jewish commentary I'm responding to here renders verse 16 as "like lions [they maul] my hands and feet" (1291). It's more likely that the passage refers to the digging out or piercing of the hands and feet rather than referring to lions. See, for example, Allen Ross, A Commentary On The Psalms, vol. 1 (Grand Rapids, Michigan: Kregel Publications, 2011), 523-4, n. 9. And see Ross' comments in note 35 on page 542. Adding another reference to lions in verse 16 disrupts the symmetry of the passage (bulls, lion, dogs in verses 12-6, then dog, lion, oxen/bulls in verses 20-1).
But even if we were to accept the lions translation, we would still have to ask why hands and feet are mentioned. Lions don't normally single out people's hands and feet when they attack. Note d-d on page 1291 of the Jewish commentary cites Isaiah 38:13. That passage does mention a lion, but doesn't single out hands and feet. Rather, it mentions the breaking of "all my bones", the same sort of attack on the whole body that Daniel 6:24 also refers to. Other passages about lion attacks also suggest some sort of tearing or other damage more significant than just going after a person's hands and feet (e.g., 1 Kings 13:26, Psalm 7:2, Jeremiah 5:6, Ezekiel 19:6, 22:25). Placing a reference to lions in Psalm 22:16 doesn't explain the most significant part of the passage, the singling out of hands and feet. Jesus' crucifixion best explains the passage under either translation.
The psalm concludes by referring to the major significance of what's happened, how people across the world will hear about it and turn to God as a result of what's been accomplished (verses 27-31). Jesus' crucifixion has had that result. It hasn't been accomplished by David's recovery from an illness that isn't referred to anywhere in the records of David's life.
The Jewish commentary I've cited above notes the similarities between Psalm 22:6-7 and Isaiah 53 (1290). The whole psalm is similar to Isaiah 53 in that both passages are best explained as prophecies fulfilled by Jesus. There's not a good naturalistic explanation for either passage. Even if the passages were only secondarily about Jesus, even if the alignment between these passages and the events of Jesus' life were a typological form of prophecy fulfillment, it would be highly unlikely that Jesus' life would naturalistically line up so well with these passages. But I don't think Jesus fulfills either passage in a secondary or typological manner. Both are primarily, not just secondarily, about Jesus.