Instructions
Complete the key questline. Most player will follow this path since the rest of the ways to get in include having special skills to bypass the lock. Obtain the first quest from Nether-Stalker Khay'ji in Area 52, located in the southwestern part of the Netherstorm in the Outland. Follow the directions he gives you. After each quest, he gives you a new quest to complete. You must complete 11 quests to retrieve the key.
Pick the door open. If you are a rogue, you already have the lock-picking skill. If your lock-picking skill is 350, you can open the lock at the Arcatraz entrance without the key. If your lock-picking skill is not yet 350, level up your character until it is.
Take blacksmithing. A skilled enough blacksmith can create either a Cobalt or Titanium Skeleton Key that opens the Arcatraz door like the normal key does. Blacksmiths must have a skill of 350 to use the Cobalt Key and a skill of 430 to use the Titanium Key.
Take engineering. A player with an engineering skill of 350 can create and use an Elemental Seaforium Charge that opens the door with ease.