The famous Hilly areas of India which are known for best snowfall in months of December and January are Uttrakhand and Himachal Pradesh. In both of States you can get best snowfall view in December. In Himachal Pradesh you can experience the best Snowfall in Manali and Macleodganj.
In Uttrakhand, you can experience the view of snowfall in Munsiyari and Nainital.
You can travel these places under 4k budget.