Since the entire island is a nature reserve, it's likely that you will be denied permission, if the purpose of entering is just tourism, although usually you won't find anybody of the Norwegian immigration office in the island to refuse your entry. But if you absolutely have to get there anyway, your best bet is to try to find out when the next research expedition is scheduled to get there, and ask if you can join them. If you have a useful occupation or skill, such as arctic research biologist, research geologist, helicopter pilot, or physician, you will probably be welcome. There's been at least one case of this happening in the past, when a bunch of radio amateurs were allowed to enter the Island for a DXpedition setting up an amateur radio station there to communicate with people across the world.
By plane
A safer way is to use a helicopter starting from a ship.
By ship
There is nothing even remotely usable as harbor, although it is possible to anchor outside of it. If you are willing to put your life at risk, you might try using a light boat with outboard engine to enter. It has been known to work, but plenty of people have tried and decided it was not worth the risk.