1) Promoted to First Lieutenant (previous rank)
2) Complete the story using only your primary weapon
3) Destroy 2,000 flying enemy units
4) Annihilate 42 boss enemies
5) Finish any stage with a 100% hit ratio
6) Deploy 200 sub-weapons
7) Play with 25 different chronomes
8) Clean game time is over 10 hours
9) Hold the highest rank for 90 seconds
Most of these challenges are self-explanatory. 2000 kills, 42 boss kills and using 200 subweapons should be obvious to anyone.
2) Can be a bit of a pain and it's suggested you do it on Normal difficulty. You can't use either your subweapons nor your capsule, so basically all you can do is hold A throughout the entire Story.
5) "Hit ratio" refers to the enemies you killed. 100% means you have to kill everything in a given level, which is actually a lot easier than it can be in other games of the genre. There's comparatively few enemies in this game, so you'll probably complete this challenge without even noticing.
7) "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.
9) You can "Hold the highest rank for 90 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.