This is fairly straight forward, but you need to be sponsored before you can complete one of these challenges. You will find battle of the brands in the mountain map menus. You need to beat the high score on the leaderboards. Some challenges are fairly easy, others aren't. When you load the battleground press to display the leaderboard and check the stats of the leading player. You must have a different Sponsor and you will then need to beat their high score.
Once you do beat their score you will be placed in first place and the achievement will unlock.
The leaderboards reset every sunday at 4am so if you have any issues just wait until they have been reset. Alternatively you can progress through the mountains and own a battleground on another mountain which isn't so populated by players.