The touristic season to visit the Island of Mykonos begins each year at the first day of the Greek easter (around april) and until the end of october. Naturally the peak season is during the months of july and august.
Each time has it's charm, according to your need, and preferences you can find the perfect moment for you to visit the Island of Mykonos.
If you want to meet plenty of people and enjoy the best parties with famous Dj's and euphoric energy then August is the perfect month for you. In another way, if you like quite places and don't like crowded places then september and october are the best options for you.
Other people prefer to travel in Mykonos during spring time at the very beginning of the season, at this moment the island is not crowded, the restaurant's staffs are not too busy and you won't have to wait a lot to have your table or dishes and the reservations in the best places will be so much easier to get. The downside of the spring season is that the sea is still chill for the one who like warm water for swimming but the weather is generally sunny and warm to get tanned.