How do you remove the battery from a Mini Cooper?

In the standard MINI (not the S type) the batter is under the hood and it's a simple plastic case top to remove first. I assume therefore that you're talking about the S type.

The battery on the MINI S is under the rear seat space. Lift up the rear seat and you'll find the battery right there. However, the battery in the 2002-2006 MINI COOPER S is, indeed, behind the seats under the floor by the rear hatch. You'll have to go in through the front door, fold the seats down, and contort your body and damage your spine to lift the floor cover, loosen the battery terminals, and heave the battery out.