First and foremost, let us remember that Ibrahim was not always Muslim. He was born to Christian parents in the Republic of Venice, before being captured and enslaved by Ottoman raiders as a child. If you are studying Ottoman history, you know that kind of thing happened a lot.

As per usual, after being enslaved by the Ottomans, captured children like Ibrahim were forcibly converted to Islam. Therefore, Islam was thrust upon him by force, and he embraced it because he had to, likely not because he wanted to. That fact led a lot of his Muslim-born contemporaries to question his true devotion to Islam.

The book, Ibrahim Pasha, Grand Vizir of Suleiman the Magnificent by Hester Donaldson Jenkins, elaborates on p. 110:

Ibrahim, we recall, was born a Christian, and probably accepted Islam only formally and not from conviction. Now and then in his career his Christian predilections appear and always injure his reputation. One instance of this was the case of the infidel Cabyz, towards whom Ibrahim was accused of being over-lenient. Another illustration of lack of consideration for Moslem prejudices was when he brought home from Buda three statues taken from the royal palace and set them up in the Hippodrome. This was in defiance of the Moslem rule, observed literally, to permit the display of "no images of anything in the heaven above, the earth beneath, or the water under the earth." Although Ibrahim was supported in this act by the tolerant sultan, it brought down on his head a clamor of horror. He was spoken of as an idolator... Ibrahim became less and less careful in religious matters as his power became more assured.

Despite the sultan's relative tolerance for non-Islamic faiths, this general dubiousness towards Ibrahim's Islamic faith is noteworthy, as it casts a general uncertainty with regards to where Ibrahim's true loyalty lies.

The real issues, it seems, begin when Ibrahim forgets his place. Suleiman had armed Ibrahim with an extraordinary amount of power, and some historians claim it eventually all got to his head, leading him to disregard his ultimate inferiority to the sultan.

Here's another excerpt elaborating on that from Jenkins' book, from p. 112:

On the Persian expedition [Ibrahim] made the grave mistake of assuming the title of Serasker-Sultan. Although as von Hammer points out the title of sultan was commonly borne by small Kurdish rulers in the country in which Ibrahim then was, yet at Constantinople there was but one sultan, and to usurp his title was to lay one's self open to the charge of unlawful ambition. Moreover as Ahmed Pasha had assumed the title upon his revolt in Egypt, the association with disloyalty must have been very strong to Suleiman. There were plenty of courtiers ready to interpret his action thus in reporting to the sultan. Here was a charge that Suleiman could hardly ignore even though he might disbelieve it for a while.

Then, you have Ibrahim's feud with Iskender Chelebi, an Ottoman [defterdar] (https://en.wikipedia.org/wiki/List_of_Ottoman_Ministers_of_Finance) (which means finance secretary), which would eventually become directly responsible for Ibrahim's execution. Chelebi was envious of Ibrahim's wealth, and Ibrahim just plain didn't like Chelebi. Their quarrel escalated over time until it turned bloodthirsty.

This is elaborated on, on p. 112:

On the expedition to Persia the smouldering hatred between the two men broke into flame. When Ibrahim proposed to take the title of Serasker-Sultan, the defterdar attempted to dissuade him and thus aroused Ibrahim's resentment. There was also an ostentatious display of wealth, the defterdar and the grand vizir each attempting to send to the army a larger number of more richly equipped soldiers, and each considering the other's contribution mean. Insults were exchanged. At length Ibrahim accused the defterdar of taking money from the royal treasury, and brought witnesses against him who were probably in Ibrahim's pay. It became a war to the death between the two enemies. Ibrahim doubtless knew that if Iskender lived he himself would be sacrificed. So he accomplished the disgrace and execution of the treasurer but he did not thereby secure his own safety. Iskender Chelebi, accused of intrigues against his master, as well as mismanagement of the public funds, was hanged at Baghdad. As he went to the gallows he sent a Parthian shot at his murderer. Calling for pen and paper, he made a written statement that not only was he guilty of conspiring with the Persians but that Ibrahim was equally guilty, and that the latter had plotted to attempt Suleiman's life, lured by Persian gold. However we may doubt Iskender's honesty in making a statement that would draw down on his enemy his own fate, the Turkish sultan would be unlikely to question it, for among the Turks the testimony of a dying man or one led to execution is of very great weight. In law it outweighs that of forty ordinary witnesses.

Which brings me to Roxelana's role in all this. Roxelana was a keenly intelligent slave girl who quickly became Suleiman's favorite wife. She was also rather power-hungry and, using her intellect, she was able to manipulate and exploit the sultan into acting on her behalf. Her move against Ibrahim Pasha (which you mentioned) is given context on page 333 of World Civilizations by Philip J. Adler and Randall L. Pouwels:

[Roxelana] began [a] campaign to remove her greatest rivals for power. Ever the wily one, she effected this by eliminating Suleiman's closest advisors, most of whom she envied for the close, daily access they had to the sultan. The first victim was no less than the grand vizier himself, Ibrahim Pasha. Ibrahim had given Suleiman thirteen years of loyal service was, ironically, used against him when Roxelana and the chief treasurer conspired to convince the Sultan that Ibrahim meant to challenge him for power.

Thus, she allegedly aided Chelebi in bringing about Ibrahim's downfall, both because she was envious of Ibrahim, and because she saw him as a threat.

It seems that Roxelana's ultimate goal was to see to it that her son inherited the Ottoman throne. The existence of Suleiman's son, Mustapha, made that impossible.

Ibrahim was a supporter of Mustapha, and was thereby viewed as an extension of that obstruction. So, both he and Mustapha needed to go. (Haven't found any great sources to add here to back this part up.)

Anyway, now, if you're Suleiman, you see Ibrahim as a man of dubious Islamic faith, who may or may not be involved in a plot to murder you, while at the same time, your beloved wife and chief treasurer (Chelebi) are insisting heavily on his being guilty. And then, to really bring it home, Suleiman allegedly experiences a troubling vision. Referencing Jenkins' book once again, on p. 114:

Suleiman's conviction of his vizir's guilt was further strengthened, as the Turkish chronicles relate, by a vision in which the murdered defterdar appeared surrounded by a celestial halo. He reproached Suleiman for submitting to the usurpation of his grand vizir, and finally threw himself on the sultan as though to strangle him. Suleiman, once convinced of Ibrahim's guilt or of the menace he was to his power, acted secretly and silently. He did not confront his favorite with accusations nor give him a chance to exculpate himself, but disposed of him swiftly.

And thus, on March 15 of 1536, Ibrahim Pasha, at age 43, was strangled to death without explanation by mute executioners at a dinner party on Suleiman's orders.

And just to wrap things up, in 1553, Suleiman had his son, Mustapha, executed. Therefore, Roxelana's son, Selim II, would successfully inherit the Ottoman throne in 1566.

EDIT: Check out Chamboz's reply below. As he says, my comment largely reflects an older, outdated account of how things went down. He elaborates on the core reasons absent from my answer, and makes some corrections.