According to Catholicism, in what cases is Purgatory necessary? As I understand it, there are at least two "steps" necessary for a person to reach Heaven; I hope this description is valid from nearly all Christian viewpoints. First the person has to choose to follow Jesus, "believe". While willing to change in order to follow, there is not yet much outward evidence that he/she is a new person. Second, Christ is the one who changes the person (with the person's cooperation), and that is usually a gradual process.

Catholic doctrine accepts Purgatory as a place one may go to after death; as I understand it the primary purpose is to complete the change, so the person is ready for Heaven.

According to this answer, not only the thief on the cross but apparently many who are preparing for baptism or recently baptized when they die, have no need to spend time in purgatory.

In essence, Christ's "fishing strategy" seems to be:


Catch a fish (don't clean it first).
"Clean": sancify, make Holy, prepare for Heaven.
Harvest: Either at death or rapture, the person enters God's realm.
Purgatory: If necessary, complete the process of cleaning.
Process complete: The person is now a full citizen of Heaven.


Yet apparently for the thief on the cross and some new believers who die while awaiting or shortly after baptism, there is no need for the "cleaning" process, either on Earth or in Purgatory. Can someone explain the Roman Catholic view of this? 

I hope my somewhat crude description is not offensive. I will change the language if it is a problem.
Very good question that I would like to see if I can answer to your satisfaction or at least give you some pointers to where to obtain a deeper understanding of the mystery of what happens to our soul after death. You can find biblical passages in my references that support this understanding by the Church and has been expanded and deepened over the centuries, but nevertheless, for sure, nobody knows, only God.

Purgatory is not only necessary for the reasons you list but also is necessary, but not limited to, erasing from our soul, not the sin per se, but the tendency for us to sin. Since we would go to purgatory "after" our personal judgement with Christ at which point we would have accepted Him or rejected Him.

This is call "Concupiscence." Here is an extract of the New Advent Catholic Encyclopedia under this subject:


  Christ by His death redeemed mankind from sin and its bondage. In baptism the guilt of original sin is wiped out and the soul is cleansed and justified again by the infusion of sanctifying grace. But freedom from concupiscence is not restored to man, any more than immortality; abundant grace, however, is given him, by which he may obtain the victory over rebellious sense and deserve life everlasting.


For an easy way to envision this, take this example: how come, even though Christ has forgiven us, in the case of Catholics via the Sacrament of Reconciliation or Confession or in protestant circles, by Accepting Christ as my Savior and be saved, we still sin and do bad things? It is very, very hard to work on our bad behaviors and tendencies and we see we fall again and again. I am talking here about the true at heart and honest to themselves believers, not the people that say one thing and do another or never had the intention to amend their ways.

So, purgatory is where we work away, with the Grace of God, that last remaining tendency to not do the will of God freely and instantly, with no second thoughts of what we want or what makes us feel good. It is the point where there is no distinction between our will and God's Will.

In the case of the Good Thief on the Cross, we do not know what that infusion of Grace my have been to participate and witness the agony and sacrifice of Jesus on the Cross. He may have been transformed so deeply by the experience that when Christ says to him "Today you will be with me in paradise." (Luke 23:43) his passage through Purgatory may have been just an instant or a few hours. Then again, we don't know what a day is (or the sense of today) in the language of eternity.

So, with God's infinity Mercy and Love, he has given way to ensure we have all that we would need in order stand in front of him in His Kingdom with a soul as pure and white as snow to praise and glorify him through eternity!

God bless!

Reference

Catechism of the Catholic Church 1031