Here is the full list of all 13 Barrels Up achievements.
You're ranked 1st against your friends
You scored over 1000!
You've hit a power up target!
You scored over 9000!
Managed to gain a direct hit streak of 10
You've managed to get to wave 10!
You've managed to get to wave 20!!
You've managed to get to wave 30!!!
You found the Sombrero
You hit a Freeze target
Get a streak of 7
Score over 50,000!
Blow up 4 Barrels with a single stick of dynamite