The Banu Qaynuqa Jews joined a pact with the Muslims, known as the constitution of Madina. With this pact, it was written in the event of an attack by an enemy that the Jewish tribes would join in the defense. Furthermore, in the pact it was also stated that whoever is in alliance, those who joined the pact, they would not side with anyone against the Muslims.
This is exactly what they did. They violated the pact, killed, and declared war against the Muslims. Hence, the Prophet (p) had no choice left but to confront the enemy.
Kitab Futuh al-Buldan
“It is reported that at the arrival of the Prophet in al-Madinah he wrote an agreement and made a covenant with the Jews of Yathrib. The Jews of Kainuka, however were the first to violate the covenant, and the Prophet expelled them from al-Madinah.” 
“‘The first treaty which the Messenger of God concluded with the Jews of Medina took place when he concluded a truce with the Nadir, Qurayza, and Qaynuqa in Medina, stipulating that they refrain from supporting the pagans and help the Muslims. This was the first of his treaties’ [i.e., with the Jews].” 
“When the Messenger of God arrived in Medina the Jews, all of them, were reconciled with him, and he wrote an agreement between him and them. The Prophet attached every tribe with its confederates and established a protection between himself and them. He stipulated conditions on them, among which it was stipulated that THEY WOULD NOT HELP AN ENEMY AGAINST HIM. When the Prophet overcame the companions of Badr and arrived in Medina, the Jews acted wrongfully and destroyed the agreement that was between them and the Messenger of God. …” 
Al-Tabari (838 – 923 CE), notice he states that Banu Qaynuqa “infringed the contract in-various ways”:
“The Campaign Against the Banu Qaynuqa’
Abu Ja’far (al-Tabari) says: The Messenger of God remained in Medina after his return from Badr. When he first came to Medina he had made a compact with its JEWS THAT THEY WOULD NOT AID ANYONE AGAINST HIM and that if any enemy attacked him there they would come to his aid. After the Messenger of God killed many polytheists of Quraysh at Badr, (the Jews) were envious and behaved BADLY TOWARDS HIM, saying, “Muhammad has not met anyone who is good at fighting. Had he met us, he would have had a battle which would be unlike a battle with anyone else.” THEY ALSO INFRINGED THE CONTRACT IN VARIOUS WAYS. (The History of Al-Tabari (Ta’rikh al-Rusul wa’l-muluk) – The Foundations Of The Community [Translated and annotated by W. Montgomery Watt and M. V. McDonald – State University Of New York Press, Albany, 1987], Volume VII (7), page 85)
In the above reports it shows that there was a agreement with the Prophet and the Banu Qaynuqa tribe. They violated the treaty. What was violated? Well, according to the following reports, al-Tabari and Ibn Ishaq inform us that while the Battle of Badr was going on, they took to arms i.e., they fought on the enemy’s side against the Muslims. al-Tabari:
“The Messenger of God remained in Medina after his return from Badr. When he first came to Medina he had made a compact with its Jews that they would not aid anyone against him and that if any enemy attacked him there they would come to his aid. After the Messenger of God killed many polytheists of Quraysh at Badr (the Jews) were envious and behaved badly towards him, saying, “Muhammed has not met anyone who is good at fighting. Had he met us, he would have had a battle which would be unlike a battle with anyone else.” They also infringed the contract in various ways. The Banu Qaynuqa were the first to infringe the agreement between them and the Messenger of God, they took to arms between Badr and Uhud.” 
“Asim b. Umar b. Qatada said that the banu Qaynuqa were the first of the Jews to break their agreements with the apostle and to go to war, between Badr and Uhud, and the Apostle besieged them until they surrendered unconditionally.” 
Besides the above proofs that the Banu Qaynuqa sided with the enemy against the Muslims. They also committed further crimes. We are told that a Muslim woman was in Market in Madinah and brought some goods with her to sell. Some of the men of banu Qaynuqa asked the woman to uncover her face, when she refused, one of them got hold of the woman’s skirt and tied it, so when she stood up, she was exposed immodestly and they started laughing at her. Upon this, the woman made a loud cry for help and one Muslim came rushing and killed one of them for what he did to the woman. With blood being spilled, the Banu Qaynuqa men in turn murdered the Muslim man.
With this we see that the Prophet (p) had no choice but to exile Banu Qainuqa criminals. We have to also remember, the Muslims had many enemies in their vicinity and could not give them chances when innocent Muslim lives were at stake. So, Prophet (p) carried the command of exiling them.
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 Kitab Futuh Al-Buldan, By al-Imam abu-l Abbas Ahmad Ibn-Jabir al-Baladhuri, (Translated by Philip Khuri Hitti, PH. D.), [New York – Columbia Univeristy, Longmans, Green & Co., Agents – London: P. S. King & Son, LTD. 1916], volume 1, page 33
 Did Muhammad Conclude Treaties with the Jewish Tribes Nadir, Qurayza and Qaynuqa`? by Michael Lecker
in Israel Oriental Studies 17 (1997), page 3
 The Life of Muhammad: Al-Waqidi’s Kitab Al-Maghazi [Translator – Rizwi Faizer], volume 1, page 87
 History of al-Tabari: The: The Foundation of the Community:, volume 7, page 85 – 86
 Ibn Ishaq’s Sirat Rasul Allah – The Life of Muhammad [Translated by A. Guillaume], page 363