The visa procedure to travel Tibet is a bit more work than any other places to travel. It’s not enough with your Chinese visa only, you need a special approval document (Tibet Travel Permit) from Tibet Tourism Bureau, and if you need a certain number of paper works if you are travelling another certain places; such as military restricted areas, border parts in Tibet etc.
But the Indian travellers have been categorized into 2 different types, the ordinary Indian nationals travelling from mainland China with the other foreign travellers and the other type (i.e. most of the travellers there are) that enters from Nepal as pilgrims to visit Mt. Kailash.
The ordinary Indian passport holding travellers coming to Tibet to visit places other than Ali prefecture get to proceed their tours like the usual travellers foreign travellers, but if you are traveling across Ali prefecture, you will be treated as a pilgrim even if it is a not for religious purpose. And if you are Indian origin with US passport, you will be considered non-pilgrim traveller whatever the purpose of your visit is.
So, Indian passport holders visiting Tibet on pilgrimage tour to Mt. Kailash or others tours with Ali prefecture (any place of Ali prefecture) in their itineraries have to process differently to get Tibet Visa. Only two organizations can authorize Indian pilgrimage tour in Tibet. Foreign Affairs Office of TAR (governmental) and Pilgrim Centre (non-governmental); short for Tibet-India pilgrim Reception Centre. Other organizations and travel agencies (including Explore Tibet) are not allowed to receive pilgrims. The fastest way to book your Kailash Pilgrimage tour is by contacting the pilgrimage tour agencies in Nepal and booking a tour in group.
About entering Tibet, almost all of Indian pilgrims used to enter through Nepal-Tibet border ZhangMu before tragic earthquake in 2015, Nepal. But since the whole town of ZhangMu and the border are closed after the earthquake, the number of options to enter Tibet has only become more expensive. You can enter from Purang, a border nearer to Kailash, for which you have to either trek for three days or take a helicopter service, you can also take flight from Kathmandu to Lhasa and start your pilgrimage tour in Lhasa, or you can choose from all the ways you can enter Tibet from mainland China.