Well, I'm no expert of Azores, I've just been there on holidays last year. But from what I gather the peak season for tuna is April-June, yet for any other pelagic fish the best season is when the water gets warmer, July-September (marlins, amberjacks, dolphinfish, etc). I'm not keen to try targeting tuna season, because the weather would be colder and more unpredictable, other fish not there yet and the tunas are not guaranteed neither. So I would rather go to Pico island second half of July.
I know of a captain who does bluewater charters, with a couple of larger boats. It was too expensive to hire for me alone, so haven't had experience with that crew. And I know a local spearo who has a RIB and could take us out, although not too far into the ocean.
So depending on the weather I would be happy with 3-4 bluewater day trips, and shore diving when the weather is not good to go on a boat.
I would like to travel other places as well, so if you know some great location, be it french polynesia, let me know