This week it would appear that the earliest players will be able to access the Destiny Beta will be through a pre-order of the game. Users will be conducting a full "public Beta test" of Destiny that will include "most major modes and activities featured in the game" before the retail launch of the game. The team responsible for the Beta launch have suggested that there's a limited time for this release and that it's only "while beta codes last".
Users will be able to play the Destiny Beta in the summer of 2014, while specific dates for this release have not yet been shared. At this time it appears that Beta codes are only being distributed through a pre-order of the game.
"Once you have redeemed your Beta code, your account will be approved for Beta access. At a later date, but prior to the Destiny Beta live date in 2014, you will be contacted via your email address, allowed to select your preferred console platform, and given a Destiny Beta download code." - Bungie
If you've already pre-ordered the game, but did so before October 1st, you'll be seeing a code printed on the receipt you received or the flyer you were given when you initiated purchase. This code may also have been sent to you through an email retail rewards program.
The code is 9 digits in the following format: XXX-XXX-XXX
Pre-orders made on or before September 31st may already have had a Beta Code emailed to them. Users who ordered inside or before September that do not have a code should content the retailer they purchased their pre-order from - "assuming they are a participating retailer". This is, again, according to Bungie.