Quick version
If you bought it through Steam, check the Downloadable content (DLC) option from the in-game menu to confirm it's authorized for you (it should be). In that case just start playing and it will download while you're in-game. Note: My first character didn't get the starting bonuses at the beginning (Golden Guns and Eridian Artifact) since the DLC wasn't loaded immediately on launch, though they showed up later. My second character did get the bonuses from the start.
Details
When I first started the game this morning I checked for DLC in the in-game menu. It listed "Borderlands 2 Premiere Club" as purchased but not downloaded. Steam also showed it as purchased: from the game library right-click Borderlands 2 and choose View Downloadable Content.
After hunting around for a while in B2 and Steam looking for a way to install it I decided to just start playing. Now it shows as purchased an installed, and while playing I was notified via pop-up that I have the golden key. So apparently it installed itself while I played. I didn't need an activation key; I purchased it through Steam a couple of days before launch. Also, in Steam I now have two entries: Borderlands 2, and Borderlands 2 Premiere Club. This is the same way Steam DLC has worked for me in the past.
How to get the starting bonuses
My first character didn't get the two starting bonuses initially: a class-specific Eridian Artifact and a set of golden guns. Later in the game they showed up, though by that time the golden guns were no longer valuable (the artifact is still good). This happened because the Premiere Club DLC was still downloading at the time I started my first character.
If you want to be sure of getting the bonuses either:
create a temporary dummy character, wait for the Premiere Club DLC
to install, then create your "real" character; or
wait at the in-game menu until the DLC installs before creating your character.
Fair warning: I'm not sure option 2 works since my process was different, but it's worth a try if you don't want to create a temporary character.
To get another Golden Key you can register at http://shift.gearboxsoftware.com and create a new character. Then select your old character and enter the shift code you were emailed from linking your email and Steam account to SHiFT.
Also, be sure to follow Gearbox Software on Facebook and Twitter. According to Randy Pitchford, President of Gearbox, they plan on doing code SHiFT code giveaways to add more Golden Keys via these official sources in the future. (source) They have several keys a week so far, usually a couple each day on Saturday/Sunday. They're still testing the concept according to Pitchford, so they may change how they do things.
As of October 2nd, 2012, Duval is still sending out SHiFT codes on his Twitter in an effort to test out the software that will eventually be used for the giveaways. Since he is just testing the software, this could stop at anytime though.
Best Answer
This chest holds extremely good loot for your level. If you haven't opened the chest yet, wait to open the chest for as long as possible. If you open the chest at max level you'll get great purple or magenta (e-tech) rarity weapons. If you open it too early, you'll have a very good weapon for maybe 5-10 levels, and then it will be useless.
When playing co op with friends new to the game, make sure to warn them of the chest. It's an extremely common problem that people open the chest with no idea that it's special. It does finally have a warning that it will consume your keys when you try and open it now though.
You get the keys by activating your preorder bonus or by entering Shift Codes from GearboxSoftware and Randy Pitchford (President of Gearbox) on Twitter. The promotional ECHOcasts account also posts them less often than the other two. They occasionally tweet Shift Codes which can be entered in game to get keys. They release a couple a week or so. Note that Randy and Gearbox don't post the same shift codes (though the occasionally retweet eachother) so you should follow all 3 accounts if you want all of them. Be quick, shift codes expire either based on time or by how many people redeem them.