What is the best time to do a wildlife photographing tour in Sri Lanka?

As a tropical country, it is very fair to say that Sri Lanka does not have any seasonal changes like in Europe. But Sri Lanka has two monsoon rains, Southwest monsoon (May to September) and Northeast monsoon (November to March). Most of national parks such as Yala, Bundala, Lunugamvehera, Minneriya Wilpattu are located in affected areas of Southeast monsoon.
The general tourist season in Sri Lanka is from November to March. Some national parks are busy with more jeeps specially in Yala. If you can avoid this period, you can enjoy calm and quite jeep safari experiences in parks. So the best time to do safaris in national parks are from April to the end of August. Only Yala national park will be closed in September for a month or 45 days in every year.  

Sri Lanka weather: http://www.meteo.gov.lk/