The word "perfection" is, broadly, a state of completeness and flawlessness.
Perfection is easy to earn in the first level. There are some possibilities of a slip-up, but it is easy. However, as the player progresses to levels like Cadaver Cavern and Park-n-Perish, this achievement becomes gradually harder, as tougher zombies appear. You can replay the earlier levels again with advanced plants like Shamrocks and Repeaters for an easier way to gain this achievement.
A known glitch was that if the final zombie does one bite to a plant other than Wall-nut or Hard-nut, and then got defeated, the player would still get Perfection.