The best time to visit tungnath and chandrashila is from Sep. to Nov. as the temple gates are open and you can take part in Arti during the evening hours from 6 to 7.pm onwards. the weather is amazing during these months and the snow-capped mountains are visible and you don't want to miss the panoramic view from Chandrashila top. you can check out some of the photos I have posted on my profile .the best route, in my opinion, is from Haldwani via kherna, Ranikhet, dwarahat, Karn Prayag, gopeshwar, mandal and finally chopta from where the Trek starts. Moderate Trek or 4km to tungnath and from their 1km steep trek to Chandrashila which is little tougher than the tungnath .stay their for a night to witness the mesmerising beauty of tungnath it's very affordable, plenty of options for pure food and water. I bet you won't regret this road trip.
If you are starting from dehradun , your first day will be from ddn to chopta, on day 2 you will start your trek to chandrashila, if yoy want to visit chandrashila sunrise you will need to start trek in early morning (around 3 am). On the same day you will return back to chopta after doing your trek. on day 3 you can go back to ddn, or you can also go for devriyatal trek which is near chopta only. the trek starts from sari village.
If you want to witness snow you can plan your trek in feb to march. Otherwise it is a all weather trek you will get to see amazing landscape and lush gree meadows.