It’s important here to keep in mind that when it comes to British imperial rule, it was never motivated by only one factor, and "imperialism" wasn’t carried out just by government actors. There was a host of people in the colonies with different professions and goals—not just government bureaucrats and the military, but their families, businesspeople, journalists, and, as is relevant to your question, missionaries. My answer here is going to focus on the nineteenth century, because that’s where my expertise lies.

Before the Indian Rebellion of 1857, British India was ruled not by the Crown (that is, the British government) but by the East India Colony, a private trading company that controlled India as a quasi-government. The official policy of the East India Company was not to interfere with native religious customs or to favor one religion over another.

Of course, that’s "official policy, not the reality on the ground in India. East India Company rule coincided with an increase in evangelicalism in Britain itself.

As an aside, in contemporary America, we tend to understand the term "evangelical" to mean "conservative Christian." However, in its technical sense, "evangelical" means a Christian who spreads the "good news" of the gospels. Evangelicals of nineteenth century Britain combined social activism and reform with Christian proselyzation, and did not see these as separate ideas. Their belief in Christianity and their belief in social reform were intertwined. For example, the famous British abolitionist William Wilberforce, an evangelical Christian, also was a fervent advocate for the Christianization of India.

In 1813, Wilberforce and his ilk secured a victory in the passing of the Charter Act of 1813, which both renewed the East India Company's charter and allowed Christian missionaries to enter India. And enter India missionaries did. If you’ve read the book Jane Eyre, you may or may not remember that when St. John Rivers proposes marriage to Jane, he wants her to come to India with him as a missionary's wife. Jane rejects his proposal, but St. John's ambition to play a part in the Christianization of India makes him very much of his time and place.

In fact, the number of missionaries in India was arguably a precipitating factor in the Indian Rebellion of 1857, which was initially ignited over rumors that cartridges in the Bengal Army's new Enfield rifles were greased with pig and/or cow fat. The army was, at the time, mostly staffed by native Indians (called sepoys), with white Europeans as officers. The new cartridges had to be bitten before they were used; thus, if they were greased with pig and cow fat, biting them would be a violation of religious prescriptions agains eating pigs (for Muslims) and beef (for Hindus). We don’t, as far as I know, have any evidence to suggest that the rumor was true. What’s important is that people believed it and it seemed plausible. The increasing numbers of missionaries looking to Christianize India contributed to an atmosphere in which native Indians feared forced conversion and disrespect from their own religious practices.

The Rebellion was unsuccessful, and included atrocities at Lucknow and Cawnpore that ignited a horrific retribution by the British. After the Rebellion, rule of India was taken over by the Crown, and British possessions in India became the British Raj. The Raj, unlike the East India Company, saw itself as a force for civilization, and civilizing India was one of its goals. For the British in the nineteenth century, "civilization" meant "Christianity." The general assumption was that, as Indians became "civilized" by the British, they would also become Christian.

As to why this didn’t successfully make India a majority Christian nation, there are a lot of factors involved. One of them is the sheer size and diversity of India—Britain never had the people required to evangelize the whole of the subcontinent. Converting people is hard work, and usually requires knowledge of the people you’re trying to convert's language and customs. Secondly, these developments coincided with a growing secularization in Britain, so that while much government policy was undergirded by Christian thought, it was less explicitly proselytizing.

That all said, Christianity is the third most practiced religion in India. Some Indian Christian communities predate colonial contact by centuries (Keralan Christians claim to have been converted by St. Thomas the Apostle!), but other Christian communities were formed during the colonial period—the northeastern Indian state of Nagaland is predominantly Baptist, while the western state of Goa has a sizeable Catholic population that was converted by the Portuguese before British rule. Thus, there’s a huge diversity within Indian Christianity itself.

Sources:

Carson, Penelope. The East India Company and Religion, 1698-1858. Boydell, 2012. (For a comprehensive treatment of the issue up to 1858.)

Philip, Kavita. Civilizing Natures: Race, Resources, and Modernity in Colonial South India. Rutgers UP, 2003. (Particularly the chapter in Christianity.)