What is the view of Islam within the Catholic Church?

The Catholic Church considers its origins to be diabolical and classifies it as a form of paganism.

Pope Eugene IV, Council of Basel, Session 19, Sept. 7, 1434: “Moreover, we trust that with God’s help another benefit will accrue to the Christian commonwealth; because from this union, once it is established, there is hope that very many from the abominable sect of Mahomet will be converted to the Catholic faith.”

Pope Clement V, Council of Vienne, 1311-1312:

“It is an insult to the holy name and a disgrace to the Christian faith that in certain parts of the world subject to Christian princes where Saracens (i.e., The followers of Islam, also called Muslims) live, sometimes apart, sometimes intermingled with Christians, the Saracen priests, commonly called Zabazala, in their temples or mosques, in which the Saracens meet to adore the infidel Mahomet, loudly invoke and extol his name each day at certain hours from a high place… This brings disrepute on our faith and gives great scandal to the faithful. These practices cannot be tolerated without displeasing the divine majesty. We therefore, with the sacred council’s approval, strictly forbid such practices henceforth in Christian lands. We enjoin on Catholic princes, one and all.. They are to forbid expressly the public invocation of the sacrilegious name of Mahomet… Those who presume to act otherwise are to be so chastised by the princes for their irreverence, that others may be deterred from such boldness.”

St. Thomas Aquinas, Summa Theologica, Pt. II, Q. 12, A. 1, Obj. 2:

“… if anyone were to… worship at the tomb of Mahomet, he would be deemed an apostate.”

St. Alphonsus, The History of Heresies & Their Refutation, Chap. 7, Article I, On Mahometanism (Islam):

“The impious sect of Mahometanism sprung up in this century [the 7th]… Mahomet, the founder of this destroying sect… was born in Arabia, in 586… He [Mahomet] pretended to have long conversations with the Archangel Gabriel, in the cave of Hera… In the year 608, being then forty years of age, he began to give out that he was a Prophet, inspired by God, and he persuaded his relatives and domestics of this first, and then began to publicly preach in Mecca, and attack idolatry. At first, the people did not very willingly listen to him, and asked him to prove his mission by a miracle; but he told them that God sent him to preach the truth, and not to work miracles. The impostor, however, boasts of having wrought one, though ridiculous in the extreme: a piece, he says, fell off from the moon into his sleeve, and he fixed it on again; and it is said that this is the reason for the Mahometans adopting the half moon as the device of their Empire… “He composed the Koran… It is a collection of precepts, taken from the Mosaic and Christian Law, together with many of his own, and interspersed with fables and ridiculous revelations…. He professes that there is but one God; but in his Alcoran [the Koran] he relates many trivialities unworthy of the Supreme Being, and the whole work is, in fact, filled with contradictions, as I have shown in my book on the ‘Truth of the Faith.’… The Mahometan Paradise, however, is only fit for beasts; for filthy sensual pleasure is all the believer has to expect there… The Mahometans shave the head, and leave only a lock of hair on the crown, by which they hope Mahomet will take them up to Heaven, even out of Hell itself.”

Pope Callixtus III:

“I vow to… exalt the true Faith, and to extirpate the diabolical sect of the reprobate and faithless Mahomet [Islam] in the East.”

Pope Pius II from the "Address on the Occasion of the Procession of St. Andrew's Head" April 12, 1462

"Turn the anger of the Almighty against the godless Turks and Barbarians who despise Christ the Lord.....

Sept 26, 1459 Pope Pius II

In the royal city of the east, they have slain the successor of Constantine and his people, desecrated the temples of the Lord, defiled the noble church of Justinian with their Mohometan abominations. Each success, will only be a stepping stone until he has mastered all the Western Monarchs, overthrow the Christian Faith, and imposed the law of his false prophet on the whole world"

Pope Leo X - Fifth Lateran Council Session 12, Mar 16, 1517

"... the Turks and other infidels ... They treat the way of true light and salvation with complete contempt and totally unyielding blindness ..."

ope Eugene IV, Council of Basel, 1434:

“… there is hope that very many from the abominable sect of Mahomet will be converted to the Catholic faith.”

On the Feast of St. Peter and Paul, June 29th 1456 Pope Calixtus III addressed a bull to all bishops, archbishops, patriarchs exhorting them to prayers and added these special directions,

"On the first Sunday of each month, processions were to be made in every diocese in order to pray that the threatened Turkish invasion might be averted; the Missae contra Paganos was to be said"

The History of the Popes, from the Close of the Middle Ages - Ludwig Freiherr von Pastor, ‎Frederick Ignatius Antrobus

Also from "A History of the Popes"

On the following day Pius II published a Bull earnestly inviting all the European Princes to the Congress Since the Emperor Constantine had given peace to the Church she had never he said been so trampled upon as she now was by the adherents of the false prophet Mahomet the bloodthirsty hosts of the venomous dragon It was a punishment from Heaven for the sins of the nations

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Muslims have an incorrect view of God, as they do not recognize the Trinity, nor the divinity of Christ. Muhammad was not a true prophet like Moses, Isaiah, etc.

However, Muslims do worship the "same" God, and Catholics should treat Muslims with love and appreciate the good things they believe.

The most authoritative and well-recognized doctrine comes from the Catecism of the Catholic Church, 841 (Part 1, Section 2, Chapter 3, Article 9, Paragraph 3, III):

The Church's relationship with the Muslims. "The plan of salvation also includes those who acknowledge the Creator, in the first place amongst whom are the Muslims; these profess to hold the faith of Abraham, and together with us they adore the one, merciful God, mankind's judge on the last day."

The United States Conference of Catholic Bishops’ Committee for Ecumenical and Interreligious Affairs published in 2013 a detailed explanation Understanding Islam: A Guide for Catholic Educators. A few excerpts:

Muslims and Christians share many common beliefs in their worship of a single Creator God who loves creation and who commands that His most cherished creations, human beings, love Him, one another, and His creation.

Christians do not accord Muhammad the same status as the biblical prophets, but may regard him as a prophetic figure on such issues as charity and the protection of the poor, widows and orphans.

Pope Paul VI issued the Declaration on the Relation of the Church to Non-Christian Religions in 1965, which included references to Jews, Hindus, and Muslims.

The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all- powerful, the Creator of heaven and earth,(5) who has spoken to men; they take pains to submit wholeheartedly to even His inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. Though they do not acknowledge Jesus as God, they revere Him as a prophet. They also honor Mary, His virgin Mother; at times they even call on her with devotion. In addition, they await the day of judgment when God will render their deserts to all those who have been raised up from the dead. Finally, they value the moral life and worship God especially through prayer, almsgiving and fasting.

It is...the burden of the Church's preaching to proclaim the cross of Christ as the sign of God's all-embracing love and as the fountain from which every grace flows.

The Church reproves, as foreign to the mind of Christ, any discrimination against men or harassment of them because of their race, color, condition of life, or religion.

More recently, Pope Francis said in Evangelii Gaudium 252-3.

The sacred writings of Islam have retained some Christian teachings; Jesus and Mary receive profound veneration and it is admirable to see how Muslims both young and old, men and women, make time for daily prayer and faithfully take part in religious services. Many of them also have a deep conviction that their life, in its entirety, is from God and for God. They also acknowledge the need to respond to God with an ethical commitment and with mercy towards those most in need.

Your question is rather broad, but I think I have covered the most important points, with authoritative sources.