This is viewed as a royal psalm, a psalm about the king.
David composed it after deliverance from his foes according to the heading. 
Look at how he describes his relationship with God in vs. 1-2.

18:1-3 He loves the LORD

18:4-6 The crisis from which the LORD delivered him

18:7-15 The LORD's dramatic intervention to rescue His servant and judge His foes

18:16-19 The LORD drew David out of the waters

Psalm 17

17:1-2  David begs God to hear his prayer

17:3-5  The LORD has examined him and tried him

17:6-8  He begs God to protect and shield him

17:9-12  David describes the wicked and their violence to him

17:13-15.  The LORD is to bring down the wicked and he longs to see their face

Connections with Psalm 16
16:1  taking refuge in the LORD. 17:7
16:8 The mention of the LORD's right hand  17:7
16:1 Keep me. 17:8
16:8 "I will not be moved" 17:5
16:5 Portion- where is our portion

Psalm 16
Heading: A Mikhtam of David- Psalms 56-60

16:1-4 David trusts in the LORD vs. those around him who trust in other gods

16:5-6 The LORD is his portion, inheritance, cup, lot, and heritage

16:7-11 The LORD is at His right hand and He will not be shaken

16:8-11 is quoted in Acts 2:25-28, 31 by Peter and Acts 13:35-37 by Paul. Both of these speakers emphasize that these words did not have their ultimate fulfillment in David. David died and was buried and his tomb underwent decay. These words have their ultimate fulfillment in Jesus and His resurrection. 

15:1 The Question: O LORD, who may abide in Your tent? Who may dwell on Your holy hill?

15:2-5a The Answer: The character of the one who continues to be in God's presence
vs. 2 Three participles describe him positively: He walks with integrity, works righteousness, speaks truth
vs. 3 negatively he does not slander with his tongue, do evil to a neighbor, or take up a reproach against a friend
vs. 4 He demonstrates who he is by the company he keeps and he keeps his word even when it is painful.
vs. 5 He does not put out his money at interest and he does not take a bribe to strike down the innocent.

15:5c The Promise: He who does these things will not be shaken.

Psalm 14:1-3 The foolishness of living without God

14:4-6 The LORD is with the righteous

14:7 O that salvation would come out of Zion.

The Hebrew word translated "there is no" as in "there is no God" is used 4 times in verses 1-3. The first time man in saying in his heart that there is no God to call him to account. The other 3 times this Hebrew word is used in vs. 1-3 is God's assessment of man: there is none good. The word is used twice in vs. 1 and twice in vs. 3. 

Psalm 14 is virtually repeated in Psalm 53.  The small differences highlight a slightly different emphasis in these two psalms. Psalm 14  has unique phrases like the LORD is with the righteous generation  and the afflicted have made the LORD his refuge.  These unique phrases are emphasized in 14:5-6. In Psalm 53 verse 5 is unique as it speaks of scattering the bones of the wicked and it stresses God's judgment of the wicked.

Romans 3:10-12 quotes Psalm 14:1-3 and Psalm 53:1-3 and emphasizes that all have sinned and are in desperate need of His grace. 

Psalm 14 ends with longing that God would bring salvation out of Zion. This salvation is through His son Jesus Christ. 

Psalm 13:1-2 How long, O LORD??
This psalm is a classic example of a psalm of lament. 
"How long" is asked 4 times with the questions surrounding it intensifying. 

13:3-4 Rescue me now

13:5-6 I will praise the LORD
The psalms of individual lament usually end in praise and the praise at the end of the psalm answers the plea expressed earlier in the Psalm. 

Psalm 12:1-4 The wicked are consuming the godly
12:1 Help- NASB; Save- ESV
12:2-3 The wicked lips of the evil men
12:5-8  The LORD will preserve the poor and afflicted
The LORD arises the save the righteous
12:6 The LORD's words are pure in contrast to the words of the wicked in 12:2-3.
12:7 The LORD will preserve them.
12:8 sons of men- this phrase appears in both vs. 1 and vs. 8. 

Psalm 11
11:1 The security of the Psalmist 
11:2-3 The threat of the wicked
11:4 The sovereignty of God
11:5-6 The destiny of the wicked
11:7 The destiny of the righteous

What does Psalm 11 say about the LORD?
11:1 He is our refuge
11:4 He is in His holy temple
11:4-5 He tests all men
11:5-6 He will bring judgment upon the wicked
11:7 He will bless the righteous

How does Jesus fulfill the picture of an innocent sufferer in Psalm 11?
11:1 in Luke 13:31-33 Jesus was basically told to flee like in 11:1
11:2 The wicked shot at Jesus secretly- Matthew 26:4; Mark 14:1; Matthew 26:55
11:1 Jesus took refuge in God- I Peter 2:22-23

Psalm 10 and Psalm 9 may have originally been one Psalm. Even if that is not the case, there are many connections between the Psalms.  Here are some of them. 
10:1- times of trouble- 9:9
10:2 the wicked- 9:5, 16, 17; 10:2, 3, 4, 13, 15
10:4 The Hebrew word for "Seek" in 9:10, 12; 10:4, 13, 15
10:11 The claim that God has forgotten- 9:12, 18
10:12 Arise- 9:19
10:16 The LORD is king- 9:4, 7
10:16 nations- 9:5, 15, 17, 19, 19, 20
10:18 vindicate is the word "judge" in 9:8

10:1-2 Where is the LORD in the midst of trouble

10:3-11 A description of the wicked

10:12-15 A plea for God to arise and judge the wicked

10:16-18 The king has heard their cry and will defend them

Psalm 9:1-2 I will give thanks to the LORD

9:3-6 My enemy will stumble and turn back

9:7-10 The LORD has established His throne for judgment

9:11-12 Sing Praises to the LORD

9:13-16 Lift me up from death and let the nations sink down

9:17-18 The wicked nations vs. the afflicted and needy

9:19-20 Do not let man prevail

The Psalm emphasizes that the LORD will...
bring judgment upon the wicked- 9:3, 5, 6, 15-16, 17, 19-20
vindicate the righteous- 9:9, 10, 12 , 18, 18