No, the Flawless pool is not account wide. It is a limited time event which can only be accessed on one character once per week.
The following steps must be taken to access the Flawless pool:
• Purchase a consumable called Flawless Empyreal Fragment from Xur for 3 legendary shards
• Use it to open a portal at Calus's throne room in The Tower
• Defeat enemies and collect rewards until completing the challenge
Once completed, you will receive powerful gear and other rewards that your character can use. However, these benefits are not shared across all of your characters.