This is it, one of the big grinds for this game.
Among the weapons you can obtain are the 5-star or Mystic weapons. To get these however, isn't as simple as buying them in the shop. You need to complete an objective on specific maps using a specific character, namely the one the weapons is for. This means that there are 145 weapons in all, one for each character, making for a doozy of a grind.
Also, besides completing the objectives, you must also be playing on hard or chaos difficulty, and the character involved must be the one completing the objectives. This means that if you want say, Guan Yu's weapon and the objective is kill 1,000 enemies in 5 minutes, Guan Yu's gotta do all the dirty work, you can't just bring in a level 100 nonuple-promoted Himiko to clear house. This of course means that you'll be needing to level up everybody (and their weapons) as needed.
Finally, some weapons have multiple eligible stages. However, you only need to complete the objective on one of them to get the weapon, not all of them (A small mercy). When you complete the objective, you will get a notice that a treasure has appeared. If you check the battle log, it will show you exactly where the treasure spawned. Go there and you will find a stereotypical treasure chest. Smack that sucker open, then collect the weapon box. Finish the stage, then collect your weapon. A word of warning here though, the game will not count the weapon until you add it to your character's inventory, so don't accidentally throw it out on the rewards screen as you'd have to do the whole thing over again.
Also, two hints to make things easier, especially for time:
1) Get Red Hare for the sweet speed boost. To get him, play chapter 3's Battle of Fan Castle (Regular, not Redux) on Hard or Chaos with Joan of Arc, Kojiro Sasaki, Wang Yi, or Xu Huang in the party (just need one), and get 1,500 KOs. The entire party can contribute to this unlike with the weapons.
2) If you are missing time requirements due to the fact that people just won't shut up and advance the level, use the pause button. The voice clip will finish up, but not run the clock down.
And of course what you're waiting for, the big-ass list (Credit to the Koei wiki for the data):
One big note here: This list is the requirements for the Chaos Difficulty. If you are doing it on hard the requirements are a little easier. The KOs required for hard are reduced by 25%. I am unsure if there is any time bonus given for Hard.Shu:
If you notice any errors or omissions, please say so in the comments or via message and I will correct it as soon as I am able.