This just crossed my mind:

The Assassin can assassinate the Basilisk meaning the Basilisk can't do an attack roll.

But in the case of the other characters the Basilisk involves another player rolling two dice and if doubles are rolled, the character is killed.

Would the Assassin be immune from this if no dice were rolled, or would two dice still need to be rolled to determine if doubles are rolled, thus killing the Assassin?

This is a Can vs. Cannot situation. The basilisk ability to roll 2 dice is trumped by the assassinate special ability of preventing any dice to be rolled.