The Psalm is quoted by Christ on the cross. Psalm 31:5 is quoted by Jesus in Luke 23:46. 
This does not mean that the psalm speaks exclusively of Jesus since 31:10 makes a reference to the writer's "iniquity"- NASB. Some versions have words like "misery" instead of "iniquity." The word "iniquity" is based on the Hebrew text and the word "misery" is based on the Greek translation of the OT. 

Notice some of the things that David experienced that Christ also experienced. 
They laid a net to trap him- vs. 4. The slandered him and plotted to kill him- vs. 13.
David was a "man of sorrows"- vs. 10 and was forsaken by those closest to him- vs. 11.
David experienced these things and was delivered from death. Jesus experienced all these things and He was crucified, but not even death stops the promises of God. God raised Jesus and provided salvation through Him.