Dr. Jose Rizal attacked the Catholic Church's doctrines on Miracles, Purgatory, Confession, Trinity, Dual Nature of Christ, etc. through his writings.

In these two novels we find passages against Catholic dogma and morals where repeated attacks are made against the Catholic religion in general, against the possibility of miracles, against the doctrine of Purgatory, against the Sacrament of Baptism, against Confession, Communion, Holy Mass, against the doctrine of Indulgences, Church prayers, the Catechism of Christian Doctrine, sermons, sacramentals and books of piety. There are even passages casting doubts on or covering with confusion God's omnipotence, the existence of hell, the mystery of the Most Blessed Trinity, and the two natures of Christ. Similarly, we find passages which disparage divine worship , especially the veneration of images and relics, devotion to the Blessed Virgin and the Saints, the use of scapulars, cords and habits, the praying of rosaries, novenas, ejaculations and indulgenced prayers. Even vocal prayers are included, such as the Our Father, the Hail Mary, the Doxology, the Act of Contrition, and the Angelus, Mass ceremonies, baptismal and exsequial rites, worship of the Cross, the use of holy water and candles, processions, bells and even the Sacred Sunday obligations do not escape scorn. (source: www.cbcponline. net)
(4) The Catholic Church opposed to the bill requiring Filipino students to read the books of Rizal because they contain passages against the Roman Catholic Church.
Senator Claro M. Recto wanted to include Jose Rizal's Noli Me Tangere and El Filibusterismo in the reading list of college students in 1956. The Catholic Church opposed the proposal claiming it would violate freedom of conscience and religion. They said "the novels belong to the past and it would be harmful to read them because they present a false picture of conditions in the country." A priest, who was introduced in the senate committee hearing as an authority on Rizal, added that the Noli was not a patriotic book since it only contained 25 patriotic passages as opposed to 120 anti-Catholic statements. A Catholic senator argued that he cannot allow his son to read Rizal's novels for fear that the boy will lose his faith. (source: www.yehey.com/ News/article. aspx?id=221212)  
With this info that we have, it is now clear that Rizal really became the enemy of the Catholic Church because of his writings. I actually didn't know about this during the time that I was taking up the subject on Dr. Jose Rizal's Life in high school. You couldn't actually read about this part of Rizal's life in public school books. What you can read is that Rizal was merely attacking the Spanish Colonial government not the Catholic Church.
But how about the said letter of Josephine Bracken to the The China Mail publication, is it really true that the Catholic Archbishop bribed the acting Governor-General of the Philippines to ensure the death of Dr. Jose Rizal?
Well, we couldn't answer it directly with Yes. Only Josephine Bracken and the said Catholic Archbishop knew the truth. But if we will consider the snippet below, we can say that there is a possibility that it is true that the Catholic Archbishop bribed Polavieja to ensure the death of Dr. Jose Rizal.
It is here that Rizal is introduced to Luis Taviel (played by Jaime Fabregas) who has been appointed to defend him at his trial. Taviel is a Spanish officer who at first mistrusts Rizal and views him as a dangerous revolutionary. Most of the movie takes place in Rizal's prison cell and involves Taviel confronting him about his life. There are frequent flashbacks but some of them are flashbacks to his novels, so it is sometimes hard to keep the order clear. Eventually Taviel learns to respect Rizal and he decides to do his best job in defending him. But it is to no avail. The evil head of the Franciscan order in Manila arranges for a new governor to take over control of the Philippines. The new governor promptly orders a show trial where the outcome has already been decided. Rizal must die. Despite his best efforts, Taviel cannot save Rizal from his fate. The verdict is reached and the execution date is set for December 30, 1896. Taviel admits to Rizal that he is ashamed to be a Spaniard. (source: www.tomandcathymark ing.com/reviews/ jose_rizal. htm)
According to that review, the head of the Franciscan Friars arranged for a new governor that would take over Philippines. And this governor ordered a show trial that the result had already been decided which was the death of Dr. Jose Rizal.
Retraction
Some argued that Dr. Rizal wrote those anti-catholic passages on his letters, poems and novels during the times when he was still excommunicated from his Catholic faith. So, he was really an enemy of the Catholic Church of that time. But prior to his execution by firing squad, he wrote a retraction letter which in turn rejects all the things he wrote about the Roman Catholic Church.
Well, it might be true that there was a retraction letter and it is said that this retraction letter was really written by Rizal. However, the authenticity of this letter is still controversial up to this day and many believed Rizal didn't sign any retraction letter.
In 1912, Rizal's family rejected a petition by the Jesuits to rebury the famous man. Instead the honor was given to the Freemasons. On December 12, 1912, the remains of Rizal were removed from his sister's home to the Masonic Temple in the Tondo section of Manila. Led by Sinukuan Lodge No. 305, several Lodges conducted a Masonic Service over the remains. The next morning the Freemsons in full regalia marched in procession to his sister's home where Rizal's remains were turned over to the government representatives. The remains were then taken to the legislative building where government officials also held funeral services before final internment at the Luneta. There has been a controversy due to a claim by the Catholic Church that on the eve of his execution Rizal had reembraced the church. The evidence refutes the claim. During his trial no cleric came to the defense of Rizal. Church officials remained silent. Only many years after his execution when Jose Rizal became known as the "George Washington" of the Philippines did the Church make the claim. (source: calodges.org/ ncrl/RIZAL. html)
Now, if the Rizal retraction is really true, then I think he did it not because he wanted to embrace back his Catholic Faith but because of some family related reasons [Especially the promise he made to his mother. ~Edgar] . But I actually doubt if Dr. Rizal really wrote it because like him I am also an ex-Catholic and I also noticed those unbiblical doctrines that the Catholic Church upheld. And that even to my death, I will never embrace back the doctrines that I found out to be unbiblical.