Battlefield 3 ''Race'' in ME3 Multiplayer

Solution 1:

You have to log in to Origin with an account that has an activated Battlefield 3 Online Pass, and the pack is only for the PC and Xbox 360 versions of ME3. From the Mass Effect Wiki:

Battlefield 3 Pack (Free, single-use) - Unlock your Battlefield 3 Limited Edition soldier! A kit containing items with an aesthetic inspired by Battlefield 3. Requires that the player signs in to Mass Effect 3 with an Origin account linked to a Battlefield 3 Online Pass. (PC and Xbox 360 only)

The Battlefield 3 "race" is mostly a Human Soldier with the power Carnage instead of Concussive Shot. Full list of Powers at the Wiki, aside from the obvious cosmetic differences.