The history of the British in India is largely an unplanned one.

The first factory (trading post) was established in 1611. It was allowed under grant by the Mughal Empire. For the first hundred years that's largely what the British were doing in India, trading. So were the French, Dutch, and Portuguese. The coasts of India were dotted by European trading posts. By the early 18th century these posts had become significant strategic resources for the Indian Princes. Gaining access to European muskets and cannons, and denying the same to your enemies had become critical in the various wars in India. Each European power has their favored states.

So when the Seven Years War between England and France broke out, it almost immediately kicked off fighting in India. As part of that the Nawab of Bengal (allied to France) demanded that the British leave their factory in Calcutta. The British refused, bribed everyone they could find in the Nawab's court, and in 1757 the Nawab's army of about 6,000 (plus about 50 French artillery men) faced off against a mixed army of about 2,200 other Bengalis and 1,700 Brits at the Battle of Plassey. The Nawab lost and the English East India Company found itself functionally ruling Bengal as the Mughal Emperor granted them right of "collection of revenues." Imagine if today Walmart raised an army, won a battle and then was granted taxation rights for say, Louisiana.

By the end of the Seven Years War, the Company had acquired additional territories, and had greatly reduced other European influences in India. Over the next century or so the Company increasingly became a country, rather than a company, their board appointing the Governor. Relationships within India were ever increasingly complicated. The Company got involved in just about everyone's squabbles. So when Mysore tried to expand against Hydrabad and the Maratha Confederacy, the Company joined in, and walked away with the lion's share of gains. But it should always be kept in mind that this was done primarily with Indian troops.

Come 1857 and the Company was likely the most powerful entity in India. Only the Mughals and Maratha were really strong enough to compete anymore. And there was a growing unease among the Indians, the British were viewed as a foreign power. Skipping over the direct causes, this kicked off the Indian Mutiny where many of the 300,000 Muslim and Hindu troops rebelled against the company. The British army was called in and put it down. They then disbanded the Company and the British crown took direct control over their Indian territories. The war also spelled the end for the Mughals and Maratha, who had attempted to join in and push the British back. The remaining independent princes either signed agreements to become subject states or were conquered fairly quickly after the establishment of the British Raj.

In short the British mostly conquered India through a series of wars between various Indian states and gaining territory as part being on the winning side. By the time they got serious about conquering all of India, they already had a good part of it. The rest was still fragmented and never put up a unified front. The one serious attempt to throw the British out in 1857 wasn't coordinated and pitted a bunch of internal squabbling states and mutineers against what may have then been the most powerful empire in the world.