Does the Roman Catholic Church have a view on Christian perfection or entire sanctification? If so what is it? To what degree do they believe believers can experience any amount of freedom from sin in this life?
Essentially the answer is "No, for most people the believer cannot be perfectly free from sin in this lifetime. Indeed, the case is even worse. We must constantly fight against the concupiscence (bias toward sin) we inherit through original sin. Through God's grace present in the Sacraments the believer may gain forgiveness of sins, and she must aim for perfection throughout her life, but she cannot hope to achieve it for more than a relatively short amount of time, so that she must be constantly on guard against sin." Catholicism does not admit the impossibility of perfection in this lifetime, but does point out its obvious rarity and difficulty "since all have sinned and fall short of the glory of God." All is not lost, however, because any lack of perfection is rectified after death in purgatory. It's theologically inescapable that this must occur either before or after death because Rev 21:27 makes clear that nothing in any way imperfect or impure or sinful can be in God's presence.
"Entire sanctification" is not a phrase typically used in Catholic theology. It's a fair question, though, to ask "To what extent do Catholics believe Christians can be free from sin in this life?"
Certainly believers are called to strive for perfection:
All the faithful of Christ of whatever rank or status, are called to the fullness of the Christian life and to the perfection of charity.
(Lumen Gentium, section 40)
In Article 4 of the Catechism of the Catholic Church, titled "The Sacrament of Penance and Reconciliation", we find a reminder:
"You were washed, you were sanctified, you were justified in the name of the Lord Jesus Christ and in the Spirit of our God" [1 Corinthians 6:11] One must appreciate the magnitude of the gift God has given us in the sacraments of Christian initiation in order to grasp the degree to which sin is excluded for him who has "put on Christ." But the apostle John also says: "If we say we have no sin, we deceive ourselves, and the truth is not in us."
(Catechism, paragraph 425)
Catholics believe that the Virgin Mary was not only free from original sin, but entirely free from personal sin throughout her whole life. However, this was a special grace of God for her, not a result purely of her own effort:
The Fathers of the Eastern tradition call the Mother of God “the All–Holy” (Panagia) and celebrate her as "free from any stain of sin, as though fashioned by the Holy Spirit and formed as a new creature." By the grace of God Mary remained free of every personal sin her whole life long.
(Catechism, paragraph 493; emphasis added)
This special grace, called the Immaculate Conception, was given only to Mary to equip her for the unique task of bearing God in her womb. It was a special appropriation of the merits of Christ obtained on the cross and applied out of the sequence of time so that Mary, too, receives salvation only through the one door to the Father, Jesus Christ. Note that we too, obtain whatever level of perfection we do in this life through grace, specifically sanctifying grace. It is not by our own merits that we are sanctified, but rather the gifts we receive from the Holy Spirit. These gifts are imparted to us through the sacraments by the Holy Spirit from the store of merits earned by Christ on Calvary. From the Catechism of the Catholic Church:
1999 The grace of Christ is the gratuitous gift that God makes to us of his own life, infused by the Holy Spirit into our soul to heal it of sin and to sanctify it. It is the sanctifying or deifying grace received in Baptism. It is in us the source of the work of sanctification:48
Therefore if any one is in Christ, he is a new creation; the old has passed away, behold, the new has come. All this is from God, who through Christ reconciled us to himself.49
2000 Sanctifying grace is an habitual gift, a stable and supernatural disposition that perfects the soul itself to enable it to live with God, to act by his love. Habitual grace, the permanent disposition to live and act in keeping with God's call, is distinguished from actual graces which refer to God's interventions, whether at the beginning of conversion or in the course of the work of sanctification.
Concerning purgatory, Augustine said, in The City of God, that "temporary punishments are suffered by some in this life only, by others after death, by others both now and then; but all of them before that last and strictest judgment" (21:13) thus indicating that no one stays in purgatory past the general judgement. The decision of whether a soul goes to heaven, purgatory and then heaven, or hell is made at the particular judgement where an individual's life is evaluated. From the Catechism again:
1030 All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.
1031 The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned.606 The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire:607
As for certain lesser faults, we must believe that, before the Final Judgment, there is a purifying fire. He who is truth says that whoever utters blasphemy against the Holy Spirit will be pardoned neither in this age nor in the age to come. From this sentence we understand that certain offenses can be forgiven in this age, but certain others in the age to come.608
Please see IV. CHRISTIAN HOLINESS in the Catechism of the Catholic Church
From the section, all these: conformed to the image of his Son, [b]e perfect, as your heavenly Father is perfect, [t]here is no holiness without renunciation and spiritual battle, and [h]e who climbs never stops going from beginning to beginning, through beginnings that have no end. He never stops desiring what he already knows, indicate that perfection is never fully attained this side of heaven.
That one of the common prayers that Catholics recite is the Our Father, that there is a Penitential Act in the Order of Mass, that there are Sacraments post Baptism to forgive sins (especially the Sacrament of Penance/Reconciliation), are answers to the question that the Church poses: who could be brave and watchful enough to escape every wound of sin?1
This is also confirmed in Scripture, for example, by the following passages:
But he who endures to the end will be saved, and but I pommel my body and subdue it, lest after preaching to others I myself should be disqualified, and [m]y little children, I am writing this to you so that you may not sin; but if any one does sin, we have an advocate with the Father, Jesus Christ the righteous, etc.
1. cf. CCC 979.