Psalm 109 is what's called an "impreccatory" psalm. It's a psalm prayed against an enemy, someone who has hurt. Many Christians read it and wonder how it squares with Jesus' teaching to "love your enemies." We always need to read Old Testament teachings through the Cross. To wish harm upon someone who has hurt us is a very natural response. The psalmist is simply being honest. He wants justice for his hurt from someone who treated him unjustly. Interestingly, we always want justice for others but mercy for ourselves. Jesus teaches us a different way to handle this pain, a path that eliminates the tit for tat retaliation that always escalates, sometimes to tribal genocide, as is evidenced in certain parts of the world today.
However, I do believe that we can pray these kinds of prayers against the enemy behind our "enemies," the evil one, who hates us and wants to destroy us. We can adamantly stand against him and desire his total and complete destruction.
Therefore, as you read Psalm 109 today, speak its words against the enemy of your soul, the evil one, Satan. He can readily be the target of all our vitriol and anger. I really believe the Lord is fine with this kind of prayer.