Hi Sadiq, Chitkul in December is tricky on rented bike. The road gets blocked all the time and is difficult to reach. They clear the roads once a week or so and untill then you wil be stuck on either side. Using a local transport or a four wheel drive is the best suggestion. You do not need any permit to visit Chitkul. Hope this helps.
Avoid bike as it will be quite cold. Also, the roads would be covered white with snow and it would become difficult to ride on them.
There will be alot of may not be able to reach Chitkul.
I'm planning to go on 24th, have you been to chitkul recently?