In terms of outright prophecy, I think that from the original context and from the NT usage, one can discern only a somewhat limited number of the Psalms as directly prophetic of Christ. A short list might be Psalms 2, 16, 22, 68, 89, 110. I'm certain someone might object and there of course could be more. My basic point stands though. For the most part the Psalms read by themselves do not directly refer to Christ, but rather life in ancient Israel.
This being the case, does not ultimately mean that the rest cannot be read as Christological. I think that if we understand all salvation history centers on Christ then we can ultimately see all of the Psalms as Christological. For help with this, we shall use the perspective of Tyconius, who was a Donatist exegete that Augustine drew on liberally. According to Tyconius, Christ could be discerned in the Scriptures through his threefold form. His type in the form of ancient Israel, from whom he drew his flesh, his actual historical reality, and then his body the Church. Ancient Israel as God's "son" anticipated Christ typologically. In the Incarnation, God the Son incorporated Israel into his own being and made Israel's life and history true of him as well. This parallels the (yet to be developed at this point in Church history!) enhypostasis-anhypostasis Christology of the fifth ecumenical council. Israel's history become anhypostasis within the hypostasis of the Son. The Church then also has a similar relationship to Christ, that Christ does to Israel. The Church is incorporated into Christ and therefore becomes his body, being part of the "totus Christus"- the "total Christ." Hence, what one says about Israel can be said in anticipation of Christ (insofar as it serves as Christ's type), and then what one says of Christ, can also said the Church insofar as they are united to Christ by faith.
How does this relate to the Psalms? Specifically it relates to how we read them in a Christological manner, while recognizing them as the prayers of ancient Israel. BTW, my suggestion in this regard comes from my old pastor Rev. Dr. Karl Fabrizius- it is not original to me. I propose that we read the Psalms in a threefold context. First, the Psalms are the prayers of ancient Israel and should be read in that sitz im leben. Nevertheless, when the Holy Spirit inspired these prayers and songs, he anticipated that the Son would take take the flesh of Israel in time. Therefore his intention in their writing was that they might also be the prayers of Jesus. Since the Psalms were the book of prayer for Second Temple Jews, every Psalm was a prayer of Jesus who literally prayed them. Therefore, every Psalm should be understood Christologically (not just the actual prophecies of the coming of Christ) since they are all literally the prayers of Christ. Even the prayers of sin should be, since Christ bore our sins, even if he himself did not actually commit any. Lastly, since the Church is Christ's body, they must also be read as also referring to the life of the Church in union and solidarity with Christ. From this we gain a threefold context for reading them that both respects their original historical context and its relationship to Christ, the center of history.