Punta Cana is warm all year long with average highs around 80 degrees Fahrenheit. December and January are the only months you're likely to see 70 degrees—and even then, only at night. After that, it gets hotter and hotter. This is a great time to come but you should book well in advance and be prepared to pay some higher rates. The week between Christmas and New Year's is especially busy.
Official hurricane season is June through November and storms seem to get their worst and most frequent in August and September. Think hard about booking during this time because you may very well have to cut your trip short and leave should a storm threaten the area. Aside from the big hurricanes, the rainiest and most dangerous months are September and October. Some Punta Cana resorts shut down for annual renovations at some point during those months so be sure to call ahead.
These are basically just the months when it's not peak season and not hurricane season. The weather is still good, there are no hurricanes around and you'll be able to find a few good deals if you look around.