You have to travel around 1100 km so you can choose the mileage vehicle and then choose budget hotels. I think Group of 5 person will cost around 50k-60k
I doubt if anyone has even actually been there from Kolkata. It takes around 4-4.5 hours from Siliguri or NJP station to Phuentsoling. A cheaper option is to go to Hasimara station from NJP station and then take a taxi to Jaigaon. This is a border town where you have to complete the formalities before being allowed to enter Bhutan. Taxis are available from Hasimara to jaigaon. Once past the border, scores of transport options available to take you to Thimpu and elsewhere. Also all in all 5-6 day trip can be covered in a maximum of 12-15k or even lesser.
If you want to save money and keep the budget around 25-30k only, I suggest you travel to Phuentsholing via NJP and cross border from there. Buses take 5-6 hours to Thimpu from there.