We should note first of all that one of the most important people in the British Empire during this time was not a man. This was, of course, Queen Victoria. While she did not actually rule since Britain was a democracy, she was still important.
One of the most important men in Britain during this time was William Gladstone, who was Prime Minister at various times during the Scramble. There is debate as to whether he or Benjamin Disraeli was more important in British imperial ventures in Africa. Disraeli was much more important in pushing to gain empire in Africa, but he died in 1881 and therefore took no further role in the Scramble. Gladstone was ambivalent about empire, but was in power during much of the Scramble.
Outside the government, one of the most important figures was Cecil Rhodes. Rhodes was a major figure in pushing for British imperialism in Africa. He was, for example, the driving force behind British colonization of what came to be known as Rhodesia.