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.
The Vault Hunter's Relic only drops the chances of white item drops by 5% and then splits that 5% between the green and blue drops. so say if the odds of drops are:
No drop:40%, white:30%, green:15%, blue:10%, purple:5%, orange:Only on named & bosses (and often about 1% or less there) OR 3 borderlands symbols on a slot machine (about a 1 in 5,000 chance)
The Vault Hunters Relic would only make that:
No drop:40%, white:25%, green:18%, blue:12%, purple:5%, orange:Only on named & bosses (and often about 1% or less there) OR 3 borderlands symbols on a slot machine (about a 1 in 5,000 chance)
assuming those numbers were accurate (the orange is pretty close but the others I am mearly speculating) then even if it were modded to be a 30% better chance (thus eliminating white drops all together) it would still only be about: None:40%, White:0%, Green:38%, Blue:17%, Purple:5%, Orange:special.
No modding of %'s will make the Vault Hunters Relic really worth wearing once others become an option. Better to wear a relic that works to improve your strengths (such as an elemental boost on a siren or a brand-specific relic that helps boost your best weapon) or compensate for a weakness (+shield capacity or max health if you find yourself dying too easily).
Best Answer
Vault Hunter Relic is the only relic that improves rare item chance.
In Borderlands 2 you can find Basic and Legendary relics.
Basic Relics
Vitality: Increases health stats.
Stockpile: Increases your ammo storage for different types of ammo.
Protection: Improves shield capabilities and damage absorption.
Strength: Improves melee abilities.
Resistance: Limits damage you take from different types of weapons.
Tenacity: Intensifies Second Wind abilities.
Proficiency: Decreases the action skill cool down rate.
Elemental: Provides an increase to elemental weapon damage.
Aggression: Provides a boost to a particular weapon type.
Allegiance: Reserved manufacturers products improved.
Legendary Relics
Lucrative Opportunity It causes shop timers to count down faster.
Deputy’s Badge It increases “Fight For Your Life Time” by 10% for every player who has this relic equipped. Location: Showdown
Sheriff’s Badge It increases “Fight For Your Life Time” by 15% for every player who has this relic on them. However, it is very rare and so more difficult to find. Location: Sheriff (Lynchwood)
Moxxi’s Endowment It increases the amount of XP earned by killing enemies but is rare. Location: The Good The Bad And The Mordicai Mission, Moxxi’s Endowment
Vault Hunter It enhances the probability of enemy dropping rare loot. It is also rare.
The Afterburner It intensifies vehicle boost capabilities and it is rare too. Location: Positive Self-Image