Have a look at this site....
http://www.wunderground.com/global/stations/76649.html
about half way down the page you will see 'History and Almanac'- just enter the date you are interested in and it will bring up records of weather on that date.
Have been to Cozumel numerous times, from June thru Nov and the weather is basically the same, hot sunny, humid in the afternoons. Temps are usually in the high 80's. That doesn't vary much all year, they just have a wet and a dry season. June is technically in the 'wet' but I wouldn't let that concern you. What happens is that on very humid days you get brief,sometimes dramatic showers late in the day,then it's fine again.
Have a look at
http://www.cozumelinsider.com/ for lots of hints by someone who lives there.
Be sure to eat at Pancho's Backyard.