Assalamu alaykum. I know that supporting the disbelievers against the muslims is disbelief. But does that also apply to the private life of the muslim? For example: A Muslim does not like a certain group of muslims because of personal matters and harms them together with a group of disbelievers, or a Muslim is a member of a criminal gang together with disbelievers and robs certain Muslims. May Allaah reward you.

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

There is no doubt that it is absolutely prohibited for the Muslim to aid the disbeliever against the Muslims and contribute to harming Muslims. Whoever travels such an evil path is subjecting himself to grave consequences. He might very likely become their ally and friend in a way that takes him out of Islam.

However, it is not correct to make a general statement that supporting the disbelievers against the Muslims is disbelief. In fact, the ruling differs according to the surrounding circumstances and context. Allaah, The Exalted, says (what means): {O you who have believed, do not take My enemies and your enemies as allies, extending to them affection while they have disbelieved...} [Quran 60:1] Al-Qurtubi commented on this verse, saying:

“When a Muslim individual is known for frequently exposing the weaknesses of his fellow Muslims and providing that information to their enemies, he is not declared a disbeliever if his purpose for doing so is worldly and he believes that he is right in doing what he does, like what Haatib did with the intention of protecting his relatives in Makkah (he tried to inform the disbelievers of the Muslims' plans to conquer Makkah) but he did not intend to leave Islam by his action.”

Dr. ʻAbd Al-ʻAzeez Al-ʻAbd Al-Lateef wrote in his research about the nullifiers of Faith entitled Verbal and Practical Invalidators of Faith:

“Backing the disbelievers against Muslims means helping the disbelievers, defending them, and supporting them against Muslims. They join their ranks, defend them with money, weapons, and speech. Such is clear disbelief that nullifies faith. This is what some scholars call Tawalli (backing them and aiding them), which they view as more specific than the general term Muwaalaah (loving them and taking them as allies), like some of the leaders of the Salafi Da'wah in Najd, although a large group of Tafseer scholars (Quran interpreters) did not make a distinction between Tawalli and Muwaalaah... If Tawalli and Muwaalaah are synonyms, then just as Muwaalaah to disbelievers is of different kinds, including a kind that takes the Muslim out of Islam like absolute Muwaalaah with them, then Tawalli is also of different kinds ... There is the absolute Tawalli which nullifies faith entirely, and there are lesser degrees than that...” [Nawaaqidh Al-Eemaan (The Nullifiers of Faith)]

Lastly, we would like to emphasize the gravity of Takfeer (accusing a Muslim individual or group of Kufr [disbelief]); it is a serious matter that must not be taken lightly. Whenever the person's Islam is established by certainty, he cannot be declared a disbeliever except with like certainty. For more benefit, please refer to fataawa 8106 and 87963 about the conditions of Takfeer.

Allaah knows best.