You can only contract Dire AIDS from the snake creatures while you are in Canada. These snakes can be found on the world map. When a snake bites you and you haven't blocked the attack, you have a chance to get Dire AIDS. When you contract Dire AIDS, you will have an red ribbon next to your name in battle, and your party member will usually tell you that you have Dire AIDS once you've left a battle as well.
For this achievement, you must continue to live with Dire AIDS through to the end of the game. Simply contract Dire AIDs, don't cure it, and continue on through to the final battle for this achievement.