Continuing on with Churchill's A History of the English Speaking Peoples, I'm at the part where England establishes control over India. It turns out that it happened quite by accident. What follows is a very abbreviated version of my understanding of the start of the whole thing.
The British East India Company had several trading bases in India. It maintained its own small army, strictly for self-protection. Its board of directors had no interest in military adventures and only wanted to keep making a very nice profit.
When the ruler of their particular part of India died, a contested succession arose and the land was threatened with civil war, anarchy and destruction. In order to maintain order, the British East India Company supported a particular claimant and gradually backed into a greater and greater role in the country. All the while, their primary interest was order and trade.