I think you're confusing two things here:
It looks like you purchased a subscription to Xbox Live Gold. Live Gold allows you to play online against other people, and unlocks certain other features.
If you want to subscribe to Forza 4 DLC, you'd want the Forza 4 Season Pass, which costs $30 but comes 6 monthly installments of Forza 4 DLC. This is the page for the Season Pass on the Xbox Marketplace.
You don't need Live Gold in order to get the Forza 4 Season Pass, but it seems you may have been confused about what you were signing up for. If you bought Live Gold recently directly from Microsoft, and you think you made a mistake, you might be able to contact Microsoft Xbox support to have the charge refunded. If you used a prepaid code, or you signed up a while back, likely you won't be able to get anything back.
If you're sure you've got the Season Pass, you'll want to access the DLC store from within Forza 4, and not via the Xbox website or the Xbox Marketplace. Season passes on the Xbox are generally only redeemable via the store within the game.