On the boss battle of the first level, it's straightfoward. With some effort you can do a no damage.
But on the second level, near the boss fight, take the blue M block. I think it ups your fire rate for all levels.
As soon as you reach a part where you can jump back and forth to go to the other layers of background, make a save state. You will need it.
If you got both the M block upgrades from both the first level and this one, even better.
Now, as soon as you note the TWO bosses falling, fire some shots, then when you note you will take a hit, quickly jump to the another layer of the background. Remember that the bosses always land on the farthest layer, so you have a slack time to fire some shots, then jump to the other layer.
If you're lucky/skilled enough, at least one of the bosses will take the hits.
Go doing the following actions:
1) Wait for when the bosses are landing on your layer
2) When they're near landing, fire. Then, quickly jump back to the other layer.
3) Repeat every step on the order.
To jump from one layer to the other, it's down + jump button to jump from the upper/farthest layer to the closest/ belowest layer. To jump from the closest layer to the farthest/upper one, press up + jump button.
If that worked, then you destroyed the armor of one of them! But you did not defeated him. Instead, the boss you just desroyed the armor will turn into a skeletal-like robot. He will be harder to defeat, mainly if the other robot still has it's armor, or is still alive.
The skeleton-like robot will be harder to avoid, but do steps 1 to 3 and you're fine.
But now, they will be on different sides of the map if they weren't already.
Do jump from a layer to other, fire one-two shots and jump back. The hard part is that you probably already took damage on the first part of this level. When you manage to destroy the first skeleton robot (there's no order on who you destroy first), it becomes easier, because there's nobody just behind you. Follow steps 1-3 and you have a chance.
Why follow the steps?
Because you probably aren't a super-human (in real world) to have fast enough reflexes. And you don't have the combat suit (that's why you are here in the first place). So, follow the steps.
If everything gone right, then you will get both the achievement after you end the level, and on level 3 you will get the Combat Suit (suit showed on the left part of the achievement icon and press start menu). Now, done! Remember, the game will probably be even harder, but hey, at least you have the suit! Iff you want, you can simply never play the game again!