Whoa - that looks like an exciting journey to take !
There are direct trains but no direct flights for this journey. kolkata (West Bengal) to Jammu (Jammu And Kashmir) distance is 1742 km and you can get there by flight + flight in just over 7 hours and 7 minutes or by train fares starting from ₹ 612.
The weather at Jammu is varying from "nice" to "hotter than nice".