Reply from Raj:
Hamilton city is a great place to spend your vacation time. Most of the activities of the island are centered around this place. Average weather in Hamilton Bermuda is no different from rest of the island as Bermuda is really tiny (about 21 square mile in size) and the general weather conditions remain same throughout the island. However I know of several occasions where it had been raining heavily at the east end in St. George's while it remained completely dry at Hamilton City.
Now coming to your question, June falls in the summer month and is part of the peak tourist season. It's considered to be in high season mainly because weather is at its best around this time and remains so through October. The temperature in June usually varies between 73°F to 81°F. The average water temperature is around 77°F, so it's a great time for swimming and snorkeling as well.
You can bring light comfortable cotton wears during this time and enjoy the sun. However the high humidity (over 80%) can make it quite uncomfortable at times. If you plan to dine in some of the more formal restaurants, get a light jacket and shoes as well.
Some of the restaurants in Bermuda follow Dress Code
and expect men to wear jackets in the evening although many are relaxing such requirements these days.
There is no rainy season or monsoon time in Bermuda. It can rain any time but usually in short spurts. In June, on an average it rains on 8 days a month.
I suggest that you check out Bermuda Weather
. It has links giving detailed weather conditions for any month along with typical day time pictures.
Another thing for you to remember is that June is the start of the official hurricane season in Bermuda. But don't be scared. Bermuda has always been very well prepared to handle hurricane. In most cases the storms or the hurricanes get diverted away and do not strike the island. Even if they do, the severe ones are only seldom seen. Suggest you check out Bermuda Hurricane
to get complete idea and information about hurricanes in Bermuda.
Hope this helps!!