Tough achievement. It will take you a multiple attempts based on developing your skill/pure luck.
Start by smashing the vases on the right-most column. Some of these (most likely) will be zombies, the others plants. Place the plants in the correct positions to take out the zombies, depending on the direction they are facing. Destroy the zombies before moving on to the next column. Place any Squash or Wall-Nut at the end of the row. Continue moving over a column at a time. Try not to use plants in a row unless you know zombies are there, empty rows are possible and happen a surprising amount of the time. If a plant starts flashing however, plant it immediately where it is most likely to benefit you (usually in a row with a lot of unbroken vases). Use the green vases if there are too many zombies for you to handle based on your plants, they should be a back-up, not a first option.
As much as possible, try to clear a row of zombies before breaking any more vases in that row.
Gargantuars will throw their imp if hit with a Mine or Squash in the 7th, 8th, or 9th column, so try and wait for them to cross into the 6th column before hitting them. The Cherries should only be used as a last-resort, and preferably only on the later levels (10+).
1. Selecting a plant's card before planting it will stop it from disappearing, so keep one selected if you are waiting for rows to clear up. You can also touch the shovel to deselect a plant.
2. Mutli-squashing: Particularly useful when a row contains a lot of buckethead zombies. By touching vases precisely when another zombie is above it, you can get line up the zombies in the same location. You can then proceed to squash both of them at once (or use the Mine, which is slightly tougher to pull off). You can also use a Wall-Nut or any other plant to slow down the zombies as they destroy it, and then squash them once they have all gathered in the same location.
3. Clear a spot for the Lantern-plant so that it can see into the greatest possible amount of vases. Nothing is better than knowing exactly what containers contain plants, zombies. Even if you find plants with the lantern, you do not necessarily have to break them yet; for example, if you find a Squash but have not yet found the Gargantuar, wait until you find him before breaking the Squash vase.
4. Wild-clicking: Used in an effort to find a plant after you have exhausted your two green vases. Start with the right-most column, and proceed to the next column if you do not find your plant. Only do this is rows that are not currently overwhelmed with zombies. You can also try to set up a multi-squash if the situation presents itself.
5. Column-points: A general idea of how many columns a zombie can walk across while being shot at by a single pea shooter. The Snow Pea and Repeaters cut this number approximately in half, and the Mines/Squashes do approximately 15 column-points of damage.
Zombie - 3
Buckethead - 15
Gargantuar - 30
Imp - 6 Pea-shooter shots
Again, these numbers are approximate. You will develop a feel for how well your defenses can handle a certain zombie throughout your play-throughs.
And last but not least, what you should expect from the 35 vases:
6 backwards facing Repeaters,
2 Snow Peas
1 Potato Mine
1 Sun vase (25, 50, or 75)
7 Basic Zombies (6 starting on level 10)
5 Buckethead Zombies
1 Gargantuar and Imp (2 starting on level 10)
1 Jack-in-the-Box Zombie
Now that you know what to expect and some strategies for dealing with those pesky zombies, you should be able to get this achievement. There is some luck involved, especially on the later levels depending on where the Gargantuars and Jack-in-the-Box spawn, but this achievement is still doable being patient, not wasting plants, and letting zombies get destroyed before breaking more zombie-containing vases.
Sorry for the length as well. Leave any comments and I'll be sure to get back to you. Good luck!