Every officer has three or four abilities they can unlock by fulfilling certain conditions during battle. Each tier becomes progressively more difficult by adding new conditions. Generally speaking, the ability tiers go like this:
Tier One starts with "Defeat X number of soldiers/officers."
Tier Two is the same as above, but they add "Within X minutes" and/or "While maintaining X-percent of health."
Tier Three and Four combines Tier One, either one or both of the Tier Two conditions, or often "Without using any of the Musou Gauge."
You can fulfill complete the Ability quests on any level, on any difficulty. Pick any level and set it on Easy, and you can unlock them pretty easily. I found mission 3 on the Wu storyline to be pretty easy, because it has a lot of officers and a lot of soldiers that you can defeat with ease right from the very beginning.
You should note that when it says "Defeat X," that means for that officer specifically, not as a team. So if you have to "Defeat 600 soldiers" for one character, and "Defeat 300 soldiers and 3 officers" for another," you'll have to defeat the 600 soldiers with the first character, and then switch to the second to defeat the 300 soldiers and 3 officers.