How did the government of Great Britain justify the Crimean War and the suppression of the Indian Mutiny?

In both cases, Great Britain was motivated by the need to protect its empire, which at that time stretched across the globe. The Russian Empire was perceived as a clear threat to the British, not least because of its vast size. As the Ottoman Empire was thought to be a check on Russian expansionism, the British joined the Crimean War on the Ottoman side in the hope of putting paid to Russian territorial ambitions.
Had the Russians prevailed, it is likely that at some point they would have turned their attentions further East, where vast swathes of its empire bordered territories occupied by the British. It was thought necessary, then, by both the British government and by public opinion, that the Russians should be put in their place, lest they get any ideas about encroaching on British interests in Asia.
As far as the Indian Mutiny was concerned, the British government regarded its suppression as essential to the maintenance of good order in the Empire. As with the Russians in the Crimean War, had the mutineers prevailed, it would've sent out a clear message that the British Empire could be undermined with impunity. It would've given Britain's rival colonial powers encouragement that the British Empire was built on shaky foundations and was vulnerable to sudden outbreaks of violence. That being the case, it was deemed essential for the colonial authorities to put down the rebellion by all available means.

The Crimean War was a response to the Russian occupation of the Danubian principalities (modern-day Romania) in 1853. The Turks declared war against Russia, and the French and British declared war on Russia after the Russians destroyed a Turkish fleet on the Black Sea. The British justification for the war was that they had to protect Turkish sovereignty against the Russians. However, the real motivations of the British were to check Russian power and reduce the Russian imperial interest in the Middle East. In the end, Russia asked for peace terms in 1856, and the Black Sea became neutralized.
The Indian Mutiny, also called the Sepoy Mutiny of 1857, began when the Sepoys, soldiers in the Bengali Army, revolted against the use of British cartridges for their rifles because they contained pork and beef fat. The beef fat was offensive to Hindus, while the pork fat offended Muslims, and a large-scale revolt ensued that was brutally crushed by the British. Atrocities were committed on both sides.
The British justified their suppression of the mutiny, which was in part a reaction against British rule in India, by characterizing the Sepoys as hideous animals and by vilifying them. In addition, the British press portrayed both Hindus and Muslims as belonging to violent and demonic religions. The Indian massacres of British civilians were used as evidence of the Indians' demonic and brutal natures. You can read accounts of the British coverage of the revolt—coverage that never questioned the rightness of British imperialism or the supposedly inferior nature of the Indians.