It's difficult to answer this specific question because no one really had any idea at the time and we still have no idea now. The Assassins claimed they were sent by Richard the Lionheart. It could have been Saladin, or maybe Guy of Lusignan. The least likely suggestion is probably Humphrey IV of Toron (whose marriage had been annulled so his wife could marry Conrad instead).
Here's an old answer from u/J-Force that discusses Conrad's assassination:
And I talked about it in another question: Why did Rashid ad-Din Sinan absolve Richard the Lionheart?
Personally, I'd say the Assassins just did it to cause as much chaos as possible. There was no deep conspiracy. I definitely don't think it was Richard, although if I remember correctly, J-Force disagrees, haha.
Now that I think about it some more, I think our previous answers are pretty helpful, but they don’t really answer this question - so, hopefully I am permitted to add a second response here!
We don’t actually know who was responsible, but the people involved at the the time certainly had their own ideas.
First of all, it helps to know why Conrad ended up as king (this is taken from an answer I wrote the other day about why Conrad was negotiating with Saladin during the crusade). It goes all the way back to King Amalric of Jerusalem, who was married to Agnes of Courtenay and had two children, Baldwin and Sibylla. But their marriage was annulled and then Amalric married the Byzantine princess Maria Komnene, and had another daughter, Isabella.
Amalric died in 1174 and was succeeded by his son, now Baldwin IV, but he was a leper and had no children. Baldwin IV died in 1185 and was succeeded by his nephew, Baldwin V, who was the son of Sibylla and her first husband William of Montferrat. William had died before Baldwin V was even born, and now Sibylla was married to Guy of Lusignan. When Baldwin V died as a child in 1186, Sibylla became queen with Guy as her consort.
Guy’s most notable achievement was losing the Battle of Hattin and being taken prisoner by Saladin in July of 1187. Saladin recaptured almost all crusader territory except Tyre - which was defended by William of Montferrat’s brother Conrad, who just happened to arrive from Europe at the right time. When Guy was released by Saladin, Conrad wouldn’t let him or Sibylla rule from Tyre. Guy laid siege to Acre instead, but during the siege, Sibylla died, and Guy now had no legal claim to the kingdom. The kingdom passed to Sibylla’s next closest relative, her half-sister Isabella.
Isabella was married to a minor crusader noble, Humphrey IV, Lord of Toron, but some crusaders thought this was the perfect opportunity to give Conrad a claim to the throne. In November 1190, as the Third Crusade was starting to arrive, Isabella and Humphrey’s marriage was annulled and Conrad married her. Not everyone agreed with this though - Philip II of France and the French crusaders went along with it, but Richard I of England did not, he supported Guy’s claim instead. This dragged on for over a year - meanwhile Acre was recaptured by the crusaders, Philip and Richard argued constantly and Philip gave up and went home, and Saladin, Richard, and Conrad all negotiated with each other separately to try to gain the upper hand.
Finally in April 1192, Richard agreed to recognize Conrad’s claim. But on April 28, before he could be crowned, before he had even left Tyre, Conrad was assassinated.
Simple so far, right?? So let's see what contemporary authors and modern historians think about it...
All the medieval Muslim and Christian accounts agree on the basic sequence of events. Conrad was eating with the Bishop of Beauvais, and was murdered by two Assassins who may have been disguised for months as his servants. Here are a few of the main accounts from people who were either present on the crusade or had first-hand information later (there are some other accounts too but they're not as important).
According to Baha ad-Din ibn Shaddad, one of Saladin’s officers who later wrote a biography of the sultan:
“He lunched on Tuesday, the 13th of the month [of Second Rabi’, i.e. 28 April], with the bishop [of Beauvais]. As he left, two of his men attacked him with daggers. He was slight of build. They kept striking him until God hastened his soul to Hell-fire. The two men were apprehended and questioned about the affair and who had put them up to it. They replied, ‘The king of England put us up to it.’ Two persons assumed authority and held the citadel until the news reached the princes, and they took over authority and the administration of the town.” (The Rare and Excellent History of Saladin, pg. 200-201)
Baha ad-Din seems to simply accept that Richard was to blame.
Imad ad-Din al-Isfahani, Saladin’s chancellor, tells a similar story but in greater detail:
“He ate and drank, sated and solaced himself, and went out and rode his horse. Suddenly two men fell upon him like two mangy wolves and with their daggers stopped his movement and struck him down near those shops. Then one of them fled and entered a church, having put out that vile soul. The Marquis, at death’s door, but still with a flicker of life in him, said ‘Take me into the church’, and they took him in thinking that he was safe there. But when that one of the two murderers saw him, he fell on him to finish him off and struck him again, blow on blow. The Franks seized the two companions, and found that they were two apostates of the Brotherhood of Isma‘ilites. They asked them who had commanded them to commit this murder, and the assassins said it was the king of England. They also said that they had been Christians for six months and had begun a life of asceticism and purification, frequenting churches and living lives of rigorous piety. One was in service with Ibn Barzān and one with the Prince of Sidon so that they could both be close to the Marquis, ensuring his confidence in them by their constant presence. Then they seized hold of his saddle-bow and slaughtered him. They were both subjected to cruel punishments and were reduced to the depths of degradation.” (Arab Historians of the Crusades, pg. 141-142)
These two accounts must represent how the events were described at Saladin’s court immediately afterwards - Baha ad-Din and Imad ad-Din were both present with Saladin at the time. Imad ad-Din also accepts that Richard was responsible, although he also implies that Saladin wasn't, since
“The death of the Marquis in such circumstances was of little benefit to us” (pg. 142)
Ibn al-Athir also knew Saladin personally, but was writing a more general history of Islam much later in the 13th century. He reports that the instigator was actually the sultan:
“Saladin made contact with the head of the Ismâ‘îlîs in Syria, namely Sinan, and encouraged him to send someone to kill the king of England. If he killed the Marquis, he would have ten thousand dinars. They were unable to assassinate the king and Sinan did not see any advantage for them in it, [being eager] that Saladin should not have a mind untroubled by the Franks and thus be free to deal with them. He was greedy to get money, so he inclined toward killing the Marquis. He sent two men disguised as monks, who became associated with the lord of Sidon and Balian’s son the lord of Ramla. They were both with the Marquis in Tyre. The two stayed with them for six months, making a show of piety. The Marquis became acquainted with them and trusted them. On the above date1 the bishop at Tyre gave a banquet for the Marquis. He attended, ate his food and drank his wine and left. The two Bâtinîs we have mentioned leapt on him and wounded him severely. One of them fled and entered a church to hide. It chanced that the Marquis was carried there to have his wounds bound. This assassin attacked and slew him. Both Bâtinîs were killed in due course.” (The Chronicle of Ibn al-Athīr, pg. 396)
But like the other Muslim authors, he notes that
“The Franks attributed his assassination to the king of England’s instigation, for he wished to become the sole ruler of the Syrian littoral” (pg. 397)