Nineteenth-century imperialists claimed they were fulfilling a "civilizing mission" in their overseas conquests. to what extent was this claim true? how was this claim ideologically supported by scientific racism?

In the US part of imperialism is to create a better, more civilized world. It was connected to the idea of "city upon a hill" and Manifest Destiny. 

In colonies such as the Philippines, Cuba, Puerto Rico, and Guam, the US introduced school and up-to-date public facilities. A system of democracy was introduced to create self-government. 
Of course, there was an economic gain from all of these places and counters the goal of spreading culture. Resources were gained and markets created. In Hawaii, economics drove the annexation and then culture followed. 
The late 19th century held a belief in Social Darwinism. This belief held that some people were superior to others and so domination over them was expected if you were the dominate species. Most Americans also believed humans were broken into further species with each race being its own species. This means that "lesser humans" needed to be controlled for their own protection.

Cherokee  removal, part of the trail of tears, refers to the  forced  relocation between 1836 and 1839 of thecherokee  nation from  their lands ingeorgia, south carolina, north carolina, tennessee, texas, and alabama to the indian territory (present day oklahoma)  in  the thenwestern united states, and the resultant deaths