The Apostle Paul and the doctrine of forgiveness? Alexander the metalworker did me a great deal of harm. The Lord will repay him for what he has done (2 Timothy 4:14).

Forgiveness is central in Christianity with all the great churches.Jesus own words:Forgive them father for they know not what they do.

Here we see the Apostle Paul speaking about "repayment" for harm done to him.
What is the Catholic doctrine on this verse or how one is to understand judgment vs forgiveness in this verse?
In the quoted passage, the Apostle Paul is not writing about paying back Alexander, but that he trusted that justice will be administered by the Lord.  Paul was consistent with Jesus' command to exercise charity, which is defined by current Catholic teaching as:


  CCC 1822 Charity is the theological virtue by which we love God above all things for his own sake, and our neighbor as ourselves for the love of God.
  
  1823 Jesus makes charity the new commandment. By loving his own "to the end,"he makes manifest the Father's love which he receives. By loving one another, the disciples imitate the love of Jesus which they themselves receive. Whence Jesus says: "As the Father has loved me, so have I loved you; abide in my love." and again: "This is my commandment, that you love one another as I have loved you." {John 15:12}


When exploring Section III of the Catechism, an extended discourse on living the Christian life (and thus witnessing for Christ), you will find reference to a number of verses in Scripture that point back toward the exercise of Charity.  


  Ephesians 4:32
  (And) be kind to one another, compassionate, forgiving one another as God has forgiven you in Christ.
  
  Luke 17:3-4
  Take heed to yourselves: If thy brother trespass against thee, rebuke him; and if he repent, forgive him.  
  
  Colossians 3:13
  bearing with one another and forgiving one another, if one has a grievance against another; as the Lord has forgiven you, so must you also do.  
  
  Luke 6:37
  Stop judging and you will not be judged. Stop condemning and you will not be condemned. Forgive and you will be forgiven.  
  
  Matthew 18: 21-22 
  Then Peter approaching asked him, "Lord, if my brother sins against me, how often must I forgive him? As many as seven times?"  Jesus answered, "I say to you, not seven times but seventy-seven times.  


The Church does not confine itself to a single passage in Scripture when addressing the need to forgive one another.  A broad range of scriptural guidance is folded into the teaching that forgiveness is an act of Charity. 



Some bullet points from current teaching:  


  The seven spiritual works of mercy:
  
  
  Counsel the doubtful.
  Instruct the ignorant.
  Admonish sinners.
  Comfort the afflicted.   
  Forgive offenses.
  Bear wrongs patiently.
  Pray for the living and the dead.
  


Part of an explanation of the Lord's Prayer.  


  
  Why do we say “Forgive us our trespasses as we forgive those who trespass against us”?  CCC 2838-2839; 2862
  
  
  By asking God the Father to pardon us, we acknowledge before him that
  we are sinners. At the same time we proclaim his mercy because in his
  Son and through the sacraments “we have redemption, the forgiveness of
  sins” (Colossians 1:14). Still our petition will be answered only if
  we for our part have forgiven first.


The importance of forgiveness


  
  How is forgiveness possible? CCC 2840-2845; 2862 
  
  
  Mercy can penetrate our hearts only if we ourselves learn how to forgive – even our enemies.  Now even if it seems impossible for us to satisfy this requirement, the heart that offers itself to the Holy Spirit can, like Christ, love even to love’s extreme; it can turn injury into compassion and transform hurt into intercession.  Forgiveness participates in the divine mercy and is a high-point of Christian prayer.