You will find an introduction to the Psalms here.

Here is a prayer to use before reading from Gregory of Nazianus (329-389), 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 One
Psalm 12 A Psalm of Lament 
Notice the structure of this psalm.
a call for help v 1a
A lament v 1a-2
  B prayer v3
    C words of the wicked v4
        D God’s response v5
    C’  words of the devout v 6
  B’ prayer v 7
A’ lament v 8
Notice that the “center” of the psalm are God’s words. Notice also the working assumption that God is already acting.
Psalm 13 A Psalm of Lament  
The Psalm begins with a four-fold “How long?” Notice how each question addresses a different aspect of suffering or distress.  The phrase in verse 3 “Consider and answer me” could be paraphrased as “Look at me! Talk to me!” Then notice the switch to an expression of trust.
Psalm 14  This psalm is a kind of wisdom psalm. It begins by sounding like verses from the book of Proverbs. This psalm is almost identical to psalm 53. “Fools” are not silly people but are the corrupt, and are contrasted with the “wise” who fear and obey God. The psalm ends with a statement of trust in God.
Psalm 15  This psalm appears to be part of a service at the Temple, that asks and answers the question “Who may abide in your tent?” The question is asked of God. “Tent” is another way of referring to the Temple. The psalm contains ten actions, both positive and negative, which describe the worthy worshiper.
Psalm 16 A song of trust  Do you think the first verse is a cry for help about a particular situation or part of a declaration of trust? The psalmist stands in distinction to others who follow other gods. The psalm declares God is faithful to those who follow.

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.