'Ruin Barriers' are the thick goo that usually blocks your path and/or hidden areas, these barriers will also have the 'Doll' heads found on the 'Ruin Enemies' scattered about. In order to destroy one, you'll have to clear any given area of all enemies, this will suck the life form out of the barrier, and turn it into a breakable 'Grey' rock. Do this 10 times, and the achievement will be yours.
This should come naturally as you progress through the story, seeing as there's well over 10 barriers you'll have to destroy in order to proceed.
NOTE: It's believed that using the
Hobby Horse to destroy barriers is the only weapon that adds to the counter, so should you come across a barrier, make sure you pull out your sledge and smash it with that. Thanks to
Flopsy86 for bringing this to my attention.