This is certainly one of the more complicated achievements in the game. To unlock the scenarios, you have to meet certain requirements. These requirements differ on each one.
(Credit goes to xenosaga123 from GameFAQS.com)
1 and 2 - Have the three team characters for each stage at level 50 proficiency (includes Cao Cao, Liu Bei, Sun Quan, Nobunaga, Hideyoshi, Ieyasu)
3 - Clear Samurai chapter 2, Wu chapter 4, and Orochi chapter 2.
4 - Clear Samurai chapter 2.
5 - Clear Samurai chapter 3.
6 - Obtain 15 ingredients.
7 - Clear Wei chapter 3, and Shu stage 3.
8 - Collect all ingredients.
9 - Clear Wei chapter 7.
10, 19, and 23 - Initially opened.
11 - Clear Wei chapter 6, Wu chapter 6, and Sengoku chapter 4.
12 - Clear Wei chapter 4, and Wu chapter 5.
13 - Clear all story modes.
14 - Clear Wu final chapter.
15 - Clear Wei final chapter.
16 and 18 - Have the three characters in each stage at level 99 (includes Gan Ning, Kotaro, Zhang Liao, Huang Zhong, Xiahou Yuan, Inahime).
17 - Clear 20 stages in story mode.
20 - Sum of all character levels equals 1,820 or more.
21 - Clear Samurai final chapter.
22 - Clear Orochi chapter 5.
24 - Sum of all character levels equals 910 or more.
25 - Clear 30 stages in story mode.
26 - Sum of all character levels equals 3,640 or more.
27 - Sum of all character proficiency levels equals 2,730 or more.
28 - Clear Orochi final chapter.
These directions may not be completely exact, but it should be fairly close. In fact, I know that several of these are flat out wrong. However, while simply progressing through the story mode, those that are listed wrong will unlock.