Imam Hussain (AS) never accepted Moawieh’s forced caliphate and his silence was due to peace treaty between Imam Hassan (AS) and Moawieh. Like Imam Ali (AS) and Imam Hassan (AS) who were forced to tolerate the caliphs including Moawieh so that the deprived people would find out about the true nature of these forced caliphs, Imam Hussain (AS) did the same and considering the condition of the time chose to stay silent. From the phrases used by Imam Hussain (AS) to describe Yazid and his father, it can be clearly realised that Imam (AS) never accepted their rules which they tried to legalise by force.
After Moawieh died, according to the peace treaty between Imam Hassan (AS) and Moawieh, caliphate must had been transferred to Imam Hussain (AS) but before his death Moawieh had tried to turn the caliphate into a hereditary system and he introduced Yazid as a Caliph and asked all the people to declare their allegiance to Yazid which was totally against the peace treaty. This shows that Moawieh from the beginning never intended to abide by the peace treaty and wanted a hereditary rule for his own family.