Probably the hardest achievement in the game. As the description states this must be done on the hardest difficulty. The rest of the achievements can be knocked out on Penguin but for this one, you are going to have to up the difficulty. If you haven't played it, Wing King drops your lives down to 1, versus 9 for Penguin, 5 for Slick Steve, and 3 for Dragon Fly.
This achievement is much easier if you are able to get at least an A Rating on all of the bosses. If you can manage an S Rating without dying, even better. Gaining a higher rating means higher points, and will result in Extra Lives at every 500,000 points.
In addition to Extra Lives gained through points, there are also 1Ups throughout the game. The first one is roughly about half way through Level 2 shortly before the Clocktower. This one seems to spawn depending on the amount of tanks and planes that you shoot in this section. If you kill enough of both, a single plane will spawn as red amongst the wave that crosses from right to left.
The second one is in Level 3 just before Poseidon and can be grabbed by killing a series of Red Planes that swoop in from the right side.
The third is right in the beginning of Level 5. A wide V of planes will dip down the top center and quickly swoop back up again. If you are quick enough you should be able to grab all of these.
Between the 1Ups and Extra Lives for every 500,000 points, this is a very doable achievement.