It depends on the particulars of the psalm, but, generally speaking, I pray all the psalms as the voice of Christ in his church or the voice of the church in Christ. These are basically the same, but with different nuances. Sometimes the prayer is of a group (church) and sometimes of an individual (Christ).
Many of the imprecatory psalms are about injustice, so they call upon the Judge of the whole earth to awaken, to act, to deliver justice for his oppressed people. Some of them are about betrayal, so they ask God to destroy the betrayer. Of course, details vary.
The strong language of these psalms--much of it the typically hyperbolic flavor of Hebrew poetry--befits the emotional language of the psalms in general. Rather than asking God, "Please give me justice against this person who has wronged me," the poet prays, "Smash the teeth in their mouth and break their arms." Ah, for more chutzpah and gusto like that in our modern milquetoast prayers!
With some imprecatory psalms, I think the main goal is to vent in a God-approved manner, to spill our guts out before him, and to demand for him to Do Something! In other words, we turn things over to the Lord, whose will is always best. Better to lament, wail, and pray imprecatory psalms than to hunt down our enemy and smash their head, as Cain did. I wonder what might have happened if Cain had not sulked in silence and instead opened up to the God with whom he was angry?
Written by Chad Bird, 1517 Scholar