"Although all Scripture breathes the grace of God, yet sweet beyond all others is the book of Psalms" (Spurgeon).
Use in temple worship – eg the Great Hallel Psalms 113-118 sang at Passover, Pentecost, and the feast of Tabernacles.
Used in song – eg the "Songs of Ascents" Psalm 120-130 sang by the men as they journeyed to Jerusalem for the "three annual feasts": Feast of Unleavened Bread, Feast of Weeks (Pentecost), Feast of Tabernacles (Exodus 23:14, Deuteronomy 16:16).
The life of Jesus was prophesied in the Psalms, and Jesus himself used the Psalms as prayers, prewritten by the Holy Spirit of God.
The Psalms describe the sorrow, loneliness, rejection and suffering which He passed through during His life of humiliation.
The Psalms were used by Jesus to:
The law gives guidance to the outward works of man – he failed in this end. Jesus stated that it is the heart attitude that is most important (see Matthew 5). The Psalms gives the heart attitude of man, especially one of a contrite spirit: of a man who knows that his sin separates him from God – the saint in the world, his trials, sorrows, the persecutions he suffers, his dependence on God, his deliverance and much else. Hence the usefulness of the Psalms is in their ability to point straight to the seat of man – the heart. Quoted more often in the NT than any other Scripture.
However, the great message of the Psalms is prophecy, which one reason it is quoted so often in the NT – since the NT is a report on the fulfilment of the prophecies pertaining to the Messiah, prophecies pertaining to the sorrows, trials and sufferings of Israel along with future prophetic words of the glories in store for His redeemed people 
The great mystery revealed in Christ, the Church, was just that – a mystery, hence the Psalms do not speak fully of this topic. (Rom 16:15, Ephesians 1, etc, explained in Ephesians 3.)
That the Gentiles should be fellow heirs, and of the same body, and partakers of his promise in Christ by the gospel: Ephesians 3:6 (KJV)
We live in the dispensation of grace – the Psalms looked forward to that dispensation. Hence, its writings reflect this. Many have stumbled on this point and have been confused by its writings. The application of Scripture as doctrine is controlled by the divine rules given in the dispensation. Interpretation of Scripture, in terms of application to life today must be done in light of the revelation of the Holy Spirit (See Ephesians and Galatians, which expand on this point).
Not every Psalms pertains to the person of Jesus Christ, for instance Ps 38:7.
Note also the words prior to the first verse, of the "title", found in many Psalms are not often, if ever, of divine inspiration, but whose genuineness are often apparent.
Hence, the Psalms can point to the Christ we are associated with, but not our association with Him, for the mystery of the church had not been revealed.
Spurgeon’s "The Treasury of David" is a must read for any one interested in the Psalms in any detail.
 Much more, and with far excellent eloquence, Arno Gaebelein’s short introduction to the Psalms is worth investigation.