One of the most common criticisms that is hurled at Mu'awiyah by Twelvers is that by bringing out his army after the murder of 'Uthman he was clearly showing that he was looking to depose 'Ali by force, regardless of what he claimed his intentions were.
Now firstly it is important to note how the Twelvers fail to comprehend the seriousness of the murder of 'Uthman. They completely fail to acknowledge that a head of state has been assassinated by a conspiracy in his own home. This is a very big deal and anyone who knows history will understand that when the head of a large empire is killed for whatever reason it at the very least usually results in some kind of civil violence.
A famous example would be the assassination of Julius Caesar by a group of senators. This resulted in a power vacuum and Caesar's allies and family raising an army to punish the perpetrators which led to a huge civil war. I am not comparing 'Uthman to Caesar but to ignore this factor as a catalyst for civil strife is criminal in relation to a study of history.
'Ali by no fault of his own was unable to find the killers so Mu'awiyah raised an army to punish the perpetrators which in the times of 'Ali and Mu'awiyah was seen as the natural cause of action, to sit and do nothing would've made Mu'awiyah seem weak in the eyes of many as the murder of a family member with no justice being dealt is intolerable and can cause feuds to be held for a long time. The Twelvers of all people should know this considering how the injustice of the killing of al-Husayn caused them to declare Bani Umayyah cursed and Kuffaar. I am not at all implying that Mu'awiyah was blaming 'Ali or that 'Ali was in any way responsible for the Fitnah. 'Ali naturally raised his army as he saw this as obstructing the authority of the state as allowing this to happen without reminding Mu'awiyah of the power and authority of the state would also make 'Ali look like a weak ruler which is a matter of great importance, a classic example would be King Stephen I of England and the anarchy in England where because of Stephen appearing weak and unfit to rule led to his vassals into ignoring his commands and England being plunged into a state of anarchy.
Conflict at this point was almost unavoidable as neither side really could afford to back down especially with the conspirators being sadly within the ranks of 'Ali. The anti-Umayyad faction, which was responsible for 'Uthman's murder within 'Ali's circles essentially are the roots of who came to be known as the Shi'ah and the Khawarij. In many ways you could say that the Khawarij are essentially a split off from the Shi'ah as they felt 'Ali had betrayed Islam and their cause so they left and made their own sect. The Shi'ah are the ones who stayed loyal fanatically to 'Ali. Knowing this it is not easy to understand why they hate Bani Umayyah so much as this is essentially the original core difference that they have with Ahl as-Sunnah. It is no coincidence that the Shi'ah are the only people who see Mu'awiyah as the sole cause of the war as they have an 'aqidah to uphold and come already with the view that Mu'awiyah is evil.