Are enemies to be despised (Psalms) or loved (Matthew)?

I read in Psalm 139:22. 

I have nothing but hatred for them; I count them my enemies.

and in Matthew 5:44:

But I tell you, love your enemies and pray for those who persecute you,

David curses his enemies in many Psalms. In fact, I don´t know of any psalm that says something similar to "love your enemies". Some say, that David is searching for the Lord's help when persecuted and that despite his hate for his enemies, he is searching for the justice of the Lord. According to this view, it seems that after praying to the Lord, David could love his enemies. I agree on it, but I think that it is a very shallow view.

Expanding a little more Psalm 139, we are informed that David considers himself aligned to the Lord because he hates the enemies of the Lord:

Do I not hate them, O LORD, who hate You? And do I not loathe those who rise up against You? I hate them with perfect hatred; I count them my enemies. Search me, O God, and know my heart; Try me, and know my anxieties; And see if there is any wicked way in me, And lead me in the way everlasting.

Jesus must have read the Psalms many and many times. In fact, many Psalms, notoriously Psalm 22, prophetically reflects Jesus. So, are Jesus teachings pro or against Psalms?

Some people see Jesus as one who loved his enemies. In fact, Jesus prayed in favor of his enemies that were killing him in the cross. However, it doesn't mean He will send everyone to heaven.

Jesus says in Matthew 7:

21 “Not everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of heaven, but only the one who does the will of my Father who is in heaven. 22 Many will say to me on that day, ‘Lord, Lord, did we not prophesy in your name and in your name drive out demons and in your name perform many miracles?’ 23 Then I will tell them plainly, ‘I never knew you. Away from me, you evildoers!’(V)

In Matthew 7, Jesus calls 'evildoers' people who see themselves as Jesus followers. He sends these people away from him. Certainly, we can qualify these people as enemies of Jesus. They are hated by Jesus.  

Judas Iscariot is portrayed as a disciple who sees in Jesus a way to conquer what he wants: money. In John chapter 12, he uses the argument of helping the poor as a cover for more access to money.

In Psalms 41:9-10 New Century Version (NCV):

My best and truest friend, who ate at my table, has even turned against me. LORD, have mercy on me. Give me strength so I can pay them back.

I believe that Jesus could have said the very same thing against Judas as David because it is evident that some psalms are really prophetic.

In conclusion, Psalms and Matthew walk together and complement each other. The problem resides in identifying the correct enemies to be loved or to be hated. 

The enemies to be loved are the ones who, clearly, don't belong to God's people, who need God's grace and understand this lack of God's grace on their own lives. A Christian must love his enemy and has power to do so when he understands how graciously the Lord has been to himself. Let me love the one who hates me because the Lord has loved me first when I myself was his enemy. 

The enemies to be hated (at least, avoided) are the ones who should be our friends because they claim to love the Lord, but in fact, they are using the Lord as means to serve their false gods (notoriously money). 

In the last supper, the disciples questioned themselves when Jesus told them that there was one traitor among them. In some way, I must study myself: am I using the Lord? or do I serve Him? It is really difficult to know the answer based on my effort. Let the Lord answer this question and correct me before it is too late, as David said:

Search me, O God, and know my heart; Try me, and know my anxieties; And see if there is any wicked way in me, And lead me in the way everlasting.