Here's my own strategy, that might give people a little more idea of how to do this in a planned out, methodical way:
1> Kill off the leftmost two enemies right away (pygar then pooka), and move towards the left side of the screen ASAP with as little digging as you can. Use their caves as your path to the left side, using the upper left pooka cave on your way to the top.
2> Here's the idea... you want to start doing vertical passes with the least amount of horizontal passes until you're done with the verticals. So once you get to the top (the blue area) go left and then all the way down. Do a U turn and go most of the way back up until you get to the area you started going vertical first after killing the pygar.
3> Around now you'll have to start worrying about the other two pookas as they'll have escaped their caves and headed your way. Remember that you can pump them up TWICE without them exploding and use this to slow them down. It's tedious but you need to preserve your lives until the very end. Obviously, if you kill one of the remaining pookas, the level will end because the other will hightail it for the hills.
4> Here's where my strategy comes in. So you've got two vertical passes done. Go to the right and start another, then do them uniformly one after the other, keeping the pookas at bay until you reach the right side. I like to do them without breaking the walls at the bottom, because when the pookas are chasing you, it gives them less direct routes to get in your way. This is really the whole point of doing the systematic vertical THEN horizontal approach. You need them to work hard to get at you so you have more time to dig.
5> Eventually the "hurry up" music will start and they'll chase after you even faster. It gets pretty hairy after a while. Try not to lose a life until you're done with all the vertical passes. On your last vertical pass, on the far right side, from the top upper right corner,make sure you make one big horizontal pass along the blue area to the left side, as it seems to be required.
6> Now we start the horizontal passes. I like to start at the bottom and work my way up. I don't think it matters much though. The key is to NOT cut holes sideways closely for them to get at you through. So I run one across the bottom, then skip the next horizontal up, and run one back a level above the next, so when they're chasing me they don't have easy access. You want them to have to travel a long ways vertically- they never turn around or double back, so once they're going vertical you don't have to worry about them until they can come around a corner and back.
7> As you finish horizontals one by one, alternating rows, you'll start running out of ways to dodge them, so be super careful to keep inflating them every few seconds. Hopefully you saved your lives for the end, because when you get down to four rows it gets almost impossible to dodge them.
8> When done, inflate the remaining ones and go to the next level. I saw other people say you don't need to, but really, what's the harm? I didn't once, and the achievement didn't pop for me even though I cleared the whole level. I'm pretty sure you need to finish the level.
To summarize, the real strategy here is to keep the horizontal walls around as long as possible so the pookas have to work a lot harder to get at you, and once they start going down the wrong path you have some breathing room. It gets harder and harder as you near finishing, but doing every ALTERNATE horizontal row makes it a little less frustrating.
Good luck, and don't feel frustrated- this is a very hard and annoying achievement, for sure. Take a break after a while.