If you really want to visit the Titanic up close the season for the submersible that goes down to the actual wreck begins in late May - early August.
This is run by the same guy who has taken James Cameron down to the sea floor many many times.
But the price is about $50,000 per person, just so you know. They do sail out to the very spot where the Titanic sank and hover there for about 6 days. That is the only way they can get there (it is too far out for helicopters).