Dominica is not a real destination for birding. Around 170 different species are counted. In the north is the Syndicate popular and in the south the Morne Trois Pitons National Park.
The bst time is during the dry season. June may also be a good option with fewer tourists and lower rates.
The dry season last from Dec. until April/May. Anyway this is one of the wettest regions in the world with 7 liters of rain each year. The climate is humid tropical. The driest month is March and the wettest is August.