Hi Faiz, the cheapest way to travel between from Dehradun to Kasol is to take an HRTC (13 hours, Rs. 700) bus from Dehradun to Kullu. From Kullu, you can either rent a bike and ride to Kasol or take another HRTC bus to Kasol. You can also rent bikes in Kasol. There are several bike rental agencies. The one that I know of is Rahul Thakur (8988265439).
Publiy transportation- the cheapest and the best always. You can go to Chandigarh via non-volvo bus and from Chandigarh to bhuntar volvos are available and afterwards you can use public buses or taxi to kasol and manikaran. I also suggest you to go kheerganga and have their hot spring water bath.