Book a train from Chennai to New Jalpaiguri station. It is a 40-47 hours journey. (If flights available at an affordable price then flight can be an option to get rid off from this hectic journey of 44h) or you can cover it with a break journey. From New, Jalpaiguri takes a bus to Phuentsholing. Buses are run by Bhutan Government also and cost around 150 Rs. It takes around 5-6 hours to reach Phuentsholing from New Jalpaiguri. Once in Phuentsholing, you are technically in Bhutan. If you want to reach Thimphu you can take another bus for 100-200 Rs more. The buses leave early in the morning. So the net cost for traveling from Delhi to Bhutan(excluded Chennai to Delhi)comes out to be anywhere between 750- 1050 Rs depending on whether you want to go to Thimphu or just enter Bhutan and travel along the border. Good Luck and Happy Travelling!!!