The requirements for this achievement are as follows:
1) Promoted to Master Sergeant (the previous rank)
2) Destroy Siriad on normal difficulty
3) Continue 100 times
4) Survive for 200 seconds without getting hit
5) Finish any stage with an "S" (or better) rating
6) Cut down every leg of Kolobok
7) Destroy 10 enemies with their own bullets
8) Finish any stage on the highest rank
1 and 3 are self-explanatory. 2 is the last boss in story mode.
The easiest way to do 4) is to just load up story mode on normal after playing the game for a bit. I can't say that it works for certain as I can't go back and do the challenge again, but when I went for this I let the cutscenes play out to maximize the time spent unharmed. I was hit by the underwater ship boss in Story Mode (he's not in Arcade) and quit out, and that was enough to complete it.
For 5) you can just play any boss you feel comfortable with in Boss Training. Matouschka is a very easy one. It still counts if you set everything to max (which is an option in Boss Training) and just destroy him with ease.
You might be going "Hey, I do that every time!" to 6). Keep in mind that it has to be done in Arcade or Score Attack. Kolobok is the first Sentinel, a mini-boss.
7) and 8) can only be done in proper arcade mode. Deflecting bullets is not hard, but holding an A rank can be. See comments for a good tip from InvidiaViribus.