I hope you get good sleep along the way coz this might take long
There are direct trains but no direct flights for this journey. Catching a flight is the fastest way to do this journey. Direct trains are available as well. Chennai (Tamil Nadu) to Varanasi (Uttar Pradesh) distance is 1860 km and you can get there by flight + flight in just over 7 hours and 15 minutes. Going by train won't be that fast but you will get fares starting from ₹ 633.
The weather at Varanasi is varying from "nice" to "hotter than nice".