uDraw Studio Instant Artist

The trick to this is realizing that your HAND is the vulnerable part, not the business-end of the fly swatter. The enemies on the first level don't seem to hurt you, and the saucer enemies on level 2 only hurt you by exploding (if you take to long to swat them). It's the boss at the end of each level to watch out for. Since the boss shoots out beams in multiple directions, you have to find a safe spot for your hand. What I did was position the fly swatting end out to the side of the boss, where one of the beams will pass through it (just above your hand). As soon as the beam passes, rush in and swat that sucker one time. The boss will move around a bit and fire again, so be moving with him to get in position again. A few more swats, and the boss will die. Do the same thing on the level 2 boss, and the achievement is yours!

EDIT: To further clarify, your hand cannot get hit AT ALL during these two first stages. If you get hit even once, that's "losing a life" and will disqualify you from getting the achievement during that run. Keep an eye on your total number of lives. If you grab an extra life, that number will increase (which is fine). If that number decreases at all, you'll have to start over. I add this edit because sometimes I didn't notice that I got hit, but I must have because my total lives went down. When this happens, you'll have to start over.

Also, BIRDSTUDIOS suggests that you may need to back all the way out to try again, rather than using the "restart" option when you fail. Considering that some people have had trouble popping this achievement, I'd say it's worth backing all the way out to make sure you get it.

The trick to this is realizing that your HAND is the vulnerable part, not the business-end of the fly swatter. The enemies on the first level don't seem to hurt you, and the saucer enemies on level 2 only hurt you by exploding (if you take to long to swat them). It's the boss at the end of each level to watch out for. Since the boss shoots out beams in multiple directions, you have to find a safe spot for your hand. What I did was position the fly swatting end out to the side of the boss, where one of the beams will pass through it (just above your hand). As soon as the beam passes, rush in and swat that sucker one time. The boss will move around a bit and fire again, so be moving with him to get in position again. A few more swats, and the boss will die. Do the same thing on the level 2 boss, and the achievement is yours!

EDIT: To further clarify, your hand cannot get hit AT ALL during these two first stages. If you get hit even once, that's "losing a life" and will disqualify you from getting the achievement during that run. Keep an eye on your total number of lives. If you grab an extra life, that number will increase (which is fine). If that number decreases at all, you'll have to start over. I add this edit because sometimes I didn't notice that I got hit, but I must have because my total lives went down. When this happens, you'll have to start over.

Also, BIRDSTUDIOS suggests that you may need to back all the way out to try again, rather than using the "restart" option when you fail. Considering that some people have had trouble popping this achievement, I'd say it's worth backing all the way out to make sure you get it.

I found that if you let the pen just lightly glide around on the tablet, and then use the A button to smash the aliens, that it works much better. You can tell if you are pressing down too hard with the pen, because the A button will stop swatting, so just lighten the pressure. This allows you to swish the pen around very quickly.
As oneshotprez has mentioned, you only have to avoid the hand getting hit, not the flyswatter part.
Hope this is helpful.

I found that if you let the pen just lightly glide around on the tablet, and then use the A button to smash the aliens, that it works much better. You can tell if you are pressing down too hard with the pen, because the A button will stop swatting, so just lighten the pressure. This allows you to swish the pen around very quickly.
As oneshotprez has mentioned, you only have to avoid the hand getting hit, not the flyswatter part.
Hope this is helpful.

I found this really tough even using the other solutions as i was getting hit by the bosses. After nearly 2 hours i switched to using my index finger instead of the tablet pen. It worked a treat first time. Im left handed so used my left index finger and used my right thumb on the A button to swat. This also seemed to stop the problem of the pen sticking down when i wanted to swat.

Hope it helps