I found this information when i did a search on 'Ask Jeeves', I dont know how different it is for the north of the island though where you are going. All i can say is i have been at that time of year 3 times before and always had it hot and dry!
Seasonal weather for Santo Domingo, Dominican Republic
Averages for June
Average High 86?F........................ Average Morning Rel Humidity 92.0%
Average Low 76?F........................ Average Afternoon Rel Humidity 76.0%
Average Mean Temp 81?F................ Typical Sky Cover Scattered
Average Dew Point 74?F .................. Average Windspeed 5.77 mph
Average Wind Direction N