The key is to have as many stars as possible. So the lesser you die, the more enemies will appear in the stages. This video(and the other two parts of it) is really helpfull to learn where exactly to use climax:
My kill stats:
After stage 5: 333
After 10th stage: 625
Total (including stage 14 and 15): 811
(i was going left at stage 4 and from there on i always choose the right option)
So there is no trick or glitch...just try to kill everything and try to stay alive as long as possible and the sooner or later you will get this medal.
No shot downs:
Dont kill any enemy so you dont earn any stars. Therefore you will have the easiest version of all the stages.
In stage 12 i flew half circles all the way. (right-->bottom-->left...and back: left-->bottom-->right) Do those half circles slowy with your left stick so you can still react on those bomb clouds(or whatever they re called). When the red rockets appear speed up with RT (i think you already know this).
But in the end its all about beeing lucky. I got to stage 12 with 9% armor (got stupidly hit by the choppers at the end of stage 11 and even collided with the walls trying to evade their rockets) and did not get hit till the end of stage 13. So i got the medal with pure luck.