Hi Sruthi,
You can rent bike from Siliguri 15 kms from Bagdogra.
Nathula pass permit can be obtained from the Tourism and Civil Aviation Department. To make your tour easy, we recommend applying through a registered Sikkim tourism travel agency. One will need a valid ID proof and two passport-sized photographs.
Here is an article that might help you with understanding the route for a bike trip - https://www.tripoto.com/nepal/trips/nepal-sikkim-motor-bike-ride-2013-6501
You can also find evrythign you need to plana trip to Sikkim here - https://www.tripoto.com/sikkim.
In addition you can find all the information in the state webiste as well- https://www.sikkimtourism.gov.in/Public/index