First things first, change the difficulty down to easy. While there are countless videos online detailing how to do this, here are some helpful tips that I used to complete the game without using a continue. You'll need to practice, memorize the levels, enemy apperances and their respective fire patterns. Playing through all the levels without using a continue can be difficult.
This last achievement will, without a doubt, be the final one you earn. 'Master of Metal Slug 2'Finish game without using continue