Belcher's analysis can be broken down into several different kinds of prayer/song.
This is useful in reflecting upon Christ, as the Psalms were His hymnbook. Much is gained when we reflect upon how each Psalm would resonate upon the lips of Jesus. Through Christ it is also pastorally useful to us as we pour out our hearts to the Lord.
This is the usual list of different kinds of Psalms. For a chart breaking each of the Psalms in their different categories see here.
a. Salvation history
b. Songs of Trust
a. Royal Psalms
b. Covenant Psalms
c. Songs of Zion
d. Temple Liturgy
5. Wisdom Psalms
6. Poems of the Law
This is so useful to know, not just so that we can understand the Psalms, but so that we can know who God is, who Jesus is and how we are to relate to God through Jesus.
That's going to be the outline to the sermon, I think...
Where is my hope? (Psalms is a a book about the coming of Christ)
Who is my God? (Psalms is a book about the character of God)
Who am I? (Psalms is a book written to help us speak to God)