The New Interpreter's Study Bible notes call Psalm 58 "puzzling." I would describe it as harsh. Verses 1-2 are addressed to "you rulers." Whoever the rulers are, they are unjust. Verses 3-5 describe them as snakelike. Verses 6-8 asks God to kill them. The remaining verses, 9-11, tells us that the righteous are happy that God kills them.
The heading for Psalm 59 relates it to incidents in 1 Samuel 19. This is another prayer for protection from enemies who are described as "dogs" in a refrain. The enemies in this psalm appear to be gentile nations. Verse 11 asks God not to kill them lest the people forget. The psalm ends with an expression of trust in God.
I can't find a clear reference for the events mentioned in the heading for Psalm 60. Joab, it says, killed 12,000 Edomites in the Valley of Salt. 1 Samuel 8:13 says that David, not Joab, killed 18,000 Edomites there. 1 Chronicles 18:2 attributes the same 18,000 Edomite deaths to Abishai. Let's just shrug this off as a mystery. The Psalm itself begins with a statement that God rejected Israel's armies, i.e. they were defeated in battle. Verses 4-5 are a prayer for victory. Verses 6-8 were probably spoken by a priest or prophet. They speak of God's rule over lands within Israel, and over some Gentile neighbors. Verses 9-12 are a prayer for help.
Psalm 61 is an individual prayer for help including a petition for the king.
Psalm 62 has a refrain that begins "For God alone my soul in silence waits." The first part of the psalm is a prayer of trust in the face of threatening enemies. The second part is an instruction to trust in God, not in things that do not last. Verse 11 uses a Hebrew poetic device:
Once God has spoken;
twice have I heard this:
that power belongs to God,...
That once...twice escalation is something we will see in the Proverbs.
Psalm 63 expresses the psalmist's desire for God which, it says, is quenched in prayer. Long ago I learned a song, a round, that set verses 3-4 to a simple tune:
Your lovingkindness is better than life.
Your loving kindness is better than life.
My lips will praise you.
Thus will I bless you.
I will lift up my hands unto the Lord.
I can't read this psalm without getting an earworm of that song. Aargh!
Psalm 64 is the prayer of someone who has become the victim of a smear campaign. The psalmist's adversaries shoot verbal arrows at him (verse 4). In return God will shoot them "with his arrow" (verse 7).
Psalm 65 is my kind of psalm. It expresses gratitude to God for forgiveness and thanks for nature's bounty. I find myself sometimes moved by a deep sense of thankfulness for undeserved grace.
The earworm picture was found at Spongepedia.
Next: Psalms 66-69