The basics of this trophy are fairly simple and requires little explaining. Kill 100 of each type of zombie. Just keep playing as a plant and rack up kills and the trophy will pop sooner or later.
2 quick hints to help it go faster.
1) Target Engineers and All-stars before anything else. Most people don't play as an Engineer (especially in Team Vanquish), so it is important to kill them when you can, even if it means dying right after getting the kill. All-stars are more common, however, they have 200 health, meaning that they often run away as you are hitting them or have enough time to strike back. Although you may attack them more often, you will die more often to them as well, giving the perception you've killed more.
2) Don't be afraid to steal kills. I know this is one of those "unwritten rules of gaming" but it can go a long way. For the most part you are honestly helping (as the game is ultimately a team game), the other player will be awarded 25 coins for an assist as compensation. Now I'm not saying do this on every enemy you come across, but for those rare ones listed in #1 it will help, promise.
Otherwise just keep playing the game and getting kills and the trophy will pop. It took me around 620 player kills before I obtained this trophy. You can check your progress in the stickerbook from the main menu, just page over using

until you reach the general zombie page. The # of player kills will be on the bottom of the left hand page.
One final thing about this trophy...After going through the guides on several different sites there seems to be some discrepancy over whether or not Garden Ops kills count towards this or not.
Unfortunately I have no good answer for you, and here is why.
While researching the topic I had a friend of mine who plays the game on the X-Box One to run some tests for me. This is because the Xbone has achievement tracking for most games, including this one. I had him play a round of garden ops and track if the %/progress bar went up at all while he was playing.
According to him it did. Meaning that those kills counted towards the trophy.
However, from my own personal experience, I had played well over 10 matches of garden ops (closer to 15) by myself. Several times on Crazy difficulty and a few more on easy/medium. Each of these rounds summoned at least 3 of each type of enemy if not more (including one round at least that included 10 engineers).
After all that was said in done I still needed 600+ player kills to gain the trophy. This included a large amount of time targeting engineers (last kill needed). Was I just that unlucky?
I'm not so sure.
So it is very possible those kills count, however I wouldn't rely to heavy on this. If somone is willing to get close to the trophy and exclusively play garden ops until the trophy is obtained, it would be amazing. And put an end to that question.