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