How can I access the Auction House from out of game?

Solution 1:

No, there currently exists no way to access the AH outside of Diablo 3.

From Battle.net Support Documents:

Will there be a mobile or Web-based auction house?

We're always on the lookout for opportunities to enhance the game experience and keep our community connected to our games through the Web or mobile devices. However, we do not have any plans to share along those lines at this time

http://us.battle.net/support/en/article/diablo-iii-auction-house-functionality#q21

Solution 2:

According to the official FAQ the only way to access the auction house is from the main menu inside the game.

Where do I access the auction house?

You can access the auction house from the campaign screen, prior to entering a game, by selecting the Auction House button from the left-side menu. If you are in the middle of playing a game, you will first need to leave the game and return to the main campaign screen in order to access the auction house.