When you plan your trip to India, one of your first concerns will be the weather. You definitely don't want your trip to Rajasthan or wherever you go in India, being ruined by loose weather conditions. There are of course major differences in climate between the north and the south of India. For Rajasthan and entire North-Eastern part of India, May and June are very hot months. The monsoon season usually starts by the end of June. However you can visit Rajasthan and Varanasi in July and August, as the monsoon rains are intense but short lasting.
Definitely, the best season to visit Rajasthan and India is from September to April. This naturally also is the high season for travellers. So you may want to plan your trip to India early, and avoid fully booked hotels and end up paying a higher price for the same service. If you plan your trip 6 /8 months ahead, you are usually safe and will be able book some great drivers in india and good 'value for money' accommodation.
However, if you are planning a trip to India during the summer, it is better to go to the north of the country, especially to Himachal Pradesh including Shimla, Manali or Dharamsala with the famous McleodGanj. The weather there is more favorable for the trip.
For the south of the country, including the states of Kerala and Tamil Nadu, the weather is more favorable from November to March, almost as for Rajasthan. Rains are frequent there from June to October. So it's better to avoid this period, especially if you want to make a tour in the famous backwaters of Kerala.