There are a few options for scuba diving while on charter.
You can have us organize rental gear for you, tell us what you need and it will all be delivered the morning of your charter. You are then free to go dive at will. Air fills are available all over the place. If you are on a larger catamaran, then you could also rent a compressor.
Going it alone does not get you to the secret spots that only local knowledge brings. If you want to do some severe diving, then I would highly recommend looking at a dive dedicated crewed boat. We know which ships offer diving and which ones have a keen diving crew.
Another option is rendezvous diving. That is where you arrange with a dive store to either meet you at the dive site or meet your boat elsewhere and take you to the site.
Rendez Vous Diving is a good option if the skipper is a nondriver as he can take the rest of your party elsewhere.
When you finish, the dive boat will drop you back off in your boat’s new location.
You can always walk into one of the dive stores and see what they have going on! If you are bringing your reg or computer with you and are heading to the Caribbean, make sure you pack them in a carry-on if you don’t want them to disappear mysteriously. Don’t forget your dive card.
We have run a dedicated dive crewed charter boat for 14 years in the BVI and can be a wealth of information.
There are some National Park areas throughout the world where it is illegal to dive with a local divemaster.