You asked “which island?” and “what to do?” but you didn't specified your budget or what time of year you are planning this trip and if it could also be a sailboat or a catamaran
Know that for a private trip (only you and the crew) the minimum rates in summer are generally from €3,500 per week for a 40 ft vessel with 4 cabins (2 cabins are for the hostess and the skipper)
This amount is only for hiring the boat itself, you will have to add the price for a professional skipper (around 200€/day), the hostess (around 150€/day), port taxes, final cleaning, food, drinks , gas for cooking, fuel for a week, etc.
Basically around 400€ / day in addition to the boat rental.
Many of these offers depart from an island like the link to "Archipel" proposed by janet which leaves from Paros
There are also some possibilities from Alimos Marina south of Athens such as Cavos Yachting:
https://www.cavoyachting.com/