Kenya has a fascinating history, being one of the oldest regions continuously occupied by humans. I haven’t any kind of specialization reading Kenyan history. I have read a couple of books on general African history, yet the wikipedia article on the History of Kenya itself surpasses my knowledge. I will summarize the salient points of the article in regard to the OP in case readers haven’t access to it.
Khoisan speakers and southern Nilotic language speaking groups arrived in Kenya in due order. Bantu expansion reached a high about 1000 a.d. Swahili is a Bantu language. Portuguese explorers arrived and established a presence in the 16th century. Ft. Jesus in Mombasa was an early site. Later Arabs arrived to assert power and influence.
In addition to the organic presence of several races before the British established a protectorate and then colony there, The Kenya Colony, in 1920, there was a collateral thread of the modernization and upgrade of the social and political awareness of Kenyans,
The Kenyan African Union was formed by Harry Thuku of the Kikuyu people who were politically excluded, in 1924.
Kenyans served in a branch of the British Army (King’s African Rifles) during the Second World War and were restive and egalitarian, independence minded afterward. A revolutionary movement developed during the Cold War era. The 1950s Mau Mau movement was a vicious conflict that had 15,000 African casualties. The British Colonial authority suppressed the rebellion, yet allowed a Kenyan political independence movement to grow within the system.
The United States under the Kennedy administration was supporting selective Kenyan college education in the United States. Britain wasn’t happy with the rise of a new educated Kenyan class of potential leaders
Following is a complete quote, rather than a paraphrase of the wiki article on the form of government just before independence (in response to the OP)
“In 1962, a KANU-KADU coalition government, including both Kenyatta and Ngala, was formed. The 1962 constitution established a bicameral legislature consisting of a 117-member House of Representatives and a 41-member Senate. The country was divided into 7 semi-autonomous regions, each with its own regional assembly. The quota principle of reserved seats for non-Africans was abandoned, and open elections were held in May 1963. KADU gained control of the assemblies in the Rift Valley, Coast and Western regions. KANU won majorities in the Senate and House of Representatives, and in the assemblies in the Central, Eastern and Nyanza regions.
Kenya now achieved internal self-government with Jomo Kenyatta as its first president. The British and KANU agreed, over KADU protests, to constitutional changes in October 1963 strengthening the central government.
Kenya attained independence on 1st June 1963 and was declared a republic on 12th December 1964 with Jomo Kenyatta as Head of State
In 1964 constitutional changes further centralised the government and various state organs were formed. One of the key state organs was the Central Bank of Kenyawhich was established in 1966.
The British government bought out the white settlers and they mostly left Kenya. The Indian minority dominated retail business in the cities and most towns, but was deeply distrusted by the Africans. As a result, 120,000 of the 176,000 Indians kept their old British passports rather than become citizens of an independent Kenya; large numbers left Kenya, most of them headed to Britain.”