Actually June is just the start of the monsoons, so you should be OK except if you're in Kerala. The monsoon starts at the south west coast, and then works it way into the country in a north east direction. Hence it's a while before it reaches north India, though S. W. India is the first to be hit.
The monsoon rains can be quite different from the standard rain I've seen in the UK/US (which tends to be rather light and goes on for extended periods). The Indian monsoon tends to be heavy and goes on for a long time. Not the kind of rain where you can walk around with an umbrella or a light coat. And drainage systems tend to buckle during the monsoon - which means that you often have to wade through substantial amounts of water