The challenges for this achievement:
1) Promoted to Colonel (previous rank)
2) Collect 50 score tokens with 100,000 value
3) Destroy a boss with its own bullets
4) Destroy 1,500 enemies with your sub-weapons
5) Play with every single chronome
6) Complete arcade with 4 different pilots
7) Reach the canyon without using a continue
8) Destroy every boss using only your primary weapon
Most of these are self-explanatory and cumulative.
3) You can "Destroy a boss with its own bullets" in Boss Training. Load up Matrouschka, set your abilities to max and pick the deflect capsule. Destroy the first few carts using your subweapon and just tear through the first few phases. Once you get to the bottom of the locomotive (or whatever it's supposed to be), activate your deflect shield by holding RT. When I did this, I had enough shield to just hold RT from the moment I entered the screen until the boss died, so it should be very easy. Just make sure the bullets actually bounce back at the cannons that fire them.
5) "Chronomes" are the builds available in any mode but story mode. After you pick your ship, pilot and capsule, you'll see a hex grid with numbers on them. These are the "chronomes," so it's just a weird word for build. It marks them with a different color when you've done them. You can pick a different chronome than the one you just made by hovering over a number and pressing X, which is nice for systematically getting them all done. You can use any mode but Story and you don't even need to finish a level; just load one up and quit out if you just need to grind through chronomes.
7) The canyon is the second part of Stage IV. As you enter Stage IV, you will fly through some kind of aerial base, then you dive down into the canyon. Basically just reach Stage IV without continuing and you will complete the challenge.
8) "Destroy every boss using only your primary weapon" is self-explanatory, but not as hard as it sounds, since it's cumulative. You don't need to do every boss in one run, just keep a checklist and do them one by one in different runs. This means you can't use subweapons or capsules. Unfortunately, it needs to be done in Arcade and cannot be done in Boss Training. Keep in mind that the last rank has a challenge for beating all of Arcade using only your primary weapon, so if you do that you will complete this challenge automatically.