You will find an introduction to the Psalms, here. Here is a prayer from Gregory of Nazianus (329-389) who was an early Church father.
Lord, as I read the psalms let me hear you singing. As I read your words, let me hear you speaking. As I reflect on each page, let me see your image. And as I seek to put your precepts into practice, let my heart be filled with joy. Amen
Book III Psalms 73-83 are psalms of Asaph. Psalms 84-85,87-88 are from the guild of Korah. Psalm 89 is from the psalms for Davidic kings.
Psalm 81 A covenant renewal liturgy similar to Psalm 50. We do not know much about covenant renewal ceremonies. Verses 1-5 are a summons to worship. Verses 6-16 reminds the consequences of disobedience. In verse 5, the reference to Joseph refers to the Northern tribes of Ephraim and Manasseh.
Beginning with verse 5b, “I hear a voice…” is the start of an oracle from God. These verses through verse 10 retell the liberation from Egypt, the time in the wilderness, and the giving of the Law. Verse 7, “the secret place of thunder” is likely Mt Sinai, where God appeared in thunder. Verses 11-16 are warnings and promises from God.
Psalm 82 A petition for justice. In the ancient near east, it was commonly believed that there was a ruling heavenly council. In this psalm tells a story of how the gods/heavenly beings lost their ruling authority. The God if Israel pronounces judgment against them and is the judge of the earth. Notice what actions and lack of actions caused the heavenly council to receive judgment.
Psalm 83 A lament, a prayer for deliverance. Verses 1 is the cry for help. Verses 2-8 describe Israel’s situation. Verses 9-18 are a prayer for victory. This psalm is not about a particular event but rather is a general call for help. The number of nations that threaten Israel, 10, is a symbolic number referring to all the enemies of Israel. This is the last of the Asaph psalms. It may also be one of the oldest psalms in the entire book of Psalms. Notice that the emphasis is not so much on the saving of Israel as it is the protection and honoring of God.
Psalm 84 This is part of a section of psalms for the guild of Korah. (42-49;84-85;87-88). Verses 1-2 are praise for the Temple. Verses 3-4 as a bird finds a home for her young, so people find rest in the Temple. We don’t actually know if people lived on the Temple grounds and it is likely that birds would not be allowed to nest on the altar. This verse is perhaps better read is metaphor. Verses 5-7 tell of the final stage of pilgrimage to the Temple. We don’t know where Baca was. The reference to “our shield” and “your anointed” are about the Davidic king who had an important role in ceremonies in Zion. Verses 10-12 extol life in the Temple.
Here are several good sources to aid your reading of the Psalms
Anderson, Bernhard W., Katheryn Pfister Darr, Understanding the Old Testament Abridged fourth Edition. (Upper Saddle River,New Jersey: Prentice Hall) 1998.
Clifford, Richard, “Psalms” in The New Oxford Annotated Bible with Apocrypha: New Revised Standard Version.Coogan, Michael D.; Brettler, Marc Z.; Perkins, Pheme; Newsom, Carol A. (2010-01-20) Oxford University Press. Kindle Edition.
Stuhlmueller, Carroll, “Psalms” in HarperCollins Bible Commentary Mays, James L. ed.(San Francisco: HarperSanFrancisco) 2000.
McCann,Jr, J. Clifton “Psalms”,in The New Interpreter’s Bible Vol 3, Keck, Leander E. ed. (Nashville: Abingdon Press) 1996.