This is the most difficult achievement in the game, as it requires you to eat one of each fruit, all in one game with only four lives. There are no continues and no stage select, so you will need to have quick reflexes and master the controls in order to achieve this. If you get Game Over, you start back at zero fruits eaten.
Fortunately, the ghosts in Pac-Man have predictable behavior, and experts developed patterns to guide Pac-Man around the maze avoiding certain death, all while getting all bonus fruits.
You can see the patterns here: http://oi62.tinypic.com/wqzus.jpg
The patterns for the first four stages (top two rows) don't work 100% for this version of the game, so it's best to freestyle those stages. If you die in any stage up to stage five (Apple 1), it's best to restart.
The bottom patterns work remarkably well for stages five and up, which is where you'll need them. You'll need to practice the path, and make all your turns flawlessly. Remember to swipe the controls in advance of the actual turn.
Once you get through all four parts of the bottom pattern, move to each corner and eat the power pellets quickly. Do your best not to die. If you do die, the pattern becomes worthless, and you'll need to improvise to finish the current stage.
With time, you should be able to get to stage 13 (Key 1), and eat a key to get the achievement.
REMINDER: Remember to keep playing after you've unlocked this achievement, as you might still need one more key for the KEY achievement.
Thanks to montrossity for suggesting the use of patterns.