1) Promoted to lieutenant (previous rank)
2) Complete the story with alternative narration
3) Absorb 200 score tokens
4) Empty 100 full capsules
5) Finish a stage unharmed
6) Fully upgrade your primary weapon
7) Clean game time is over 5 hours
8) Complete the same stage with all airplane types
9) Spend 100 seconds on the highest rank
2) In order to "complete the story with alternative narration," you must first beat the entire story mode on challenging difficulty. You can't just load up the last chapter and finish that. After unlocking alternative narration, it becomes an option when you start any story mode. You can't load up the last chapter for this either, though, and need to play through the entire story mode with the new narration on. It's not very interesting - it's just new text on the text screens and a new ending that doesn't change anything.
5) "Finish a stage unharmed" is most easily done in Story Mode on Normal, as that is the lowest difficulty available. It does not unlock in Boss Training either. Note that "unharmed" is kind of a poor choice of words; they mean that you can't get hit at all, even if you have a shield saving you.
6) Many of these challenges can be obtained in Boss Training mode. For this achievement, you can complete "Fully upgrade your primary weapon" simply be selecting to start with a maxed out weapon in Boss Training. Clarified by wariomona: this only unlocks if you get hit, drop your upgrades and pick them all back up again.
7) The same goes for "Spend 100 seconds on the highest rank" - you can just hang out on a boss you feel comfortable with dodging and hold the fire button (which raises your rank) and wait 100 seconds. Level 1 in arcade is also an easy choice.
8) "Complete the same stage with all airplane types" claims it can be done in "any game mode," but Boss Training does not count and you can't pick your ship in Story, so it has to be done in Arcade or Score Attack. The first level is the easiest, of course. This one caused some trouble for me and a friend; the game didn't seem to recognize us completing the challenge until we did it in one sitting. Might have been a fluke but if you're having trouble, try that.
9) You can "Hold the highest rank for 100 seconds" by playing through the first level and making sure you don't get hit or use your subweapon or capsule too much. The rank bar in the top left corner (in every mode but Story) will let you know what rank you're currently on, and you can raise your rank by shooting at nothing. See the How to Play section of the game for a good explanation of the ranking system. You can complete this on the first level. (Very strange how this achievement is for 100 seconds but the next achievement is for 90 seconds...)