Either go in summer (March - June) for pleasant whether or in winters (October - February) for snow. I would not recommend in monsoon as the weather is very humid and wet. Hilly regions are also precarious for driving.
It is basically depends on the area specifically in Uttrakhand where you want to go if it is cities luke Haridwar, Dehradun which are not on hills you can visit anytime but for hilly area's you need to check first the climate situation first at that time. avoid the farther hilly area's in June-July it will heavy rains during that time. thanks.