Possible connections between Psalm 57 and the background of I Samuel 24
1. The word for "wings" in Psalm 57:1 is the same word "edge" for Saul's garment in I Sam. 24:4, 5, 11, 11
2. The "sons of men" use their tongues against David in Ps. 57:4 and David asks Saul why he is listening to man (same word) who tell him David is trying to kill him.
3. They prepare a net for David's soul or life in Ps. 57:6 and Saul seeks David's life in I Sam. 24:11.
4. While Ps. 57:6 expresses the idea of Lex talionis, in I Sam. 24 Saul who is pursuing David is helpless in David's hands. 

Psalm 57 and Jesus
57:1 Compare Matt. 23:37 and Luke 13:34. 
57:5, 11 The word "exalted" translated in the Septuagint is the same word for Christ being "lifted up" on the cross- John 3:14; 8:28; 12:32
57:5, 11 God's glory is displayed in the cross- John 12:23, 28; 13:31-32; 17:1, 5
57:3, 10 God's lovingkindness and truth are pre-eminently displayed in the cross
57:9 God is praised among all nations and peoples- Romans 15:9