Clear Zhao Yun's Musou Mode.
Each achievement's criteria is the same - complete X character's Musou Mode. I'll give you a list of the missions each character will complete in their Musou Mode here. Battles are won in one of two ways: either by killing the enemy commander, or by escorting your commander to their escape point (which will be marked in yellow on the Objectives screen). I've added pertinent notes about certain missions, but for a more detailed walkthrough of the game, please refer to a Dynasty Warriors 5 FAQ. I'd recommend Rampidzier's FAQ.
1 - Ji Province
2 - Hu Lao Gate: Lu Bu will appear towards the end of the stage; go around to the north and defeat Zhang Liao to bypass him.
3 - Chang Ban: Liu Bei must reach the escape point, but he won't move until the boats arrive. You can wait for him, or wait until Cao Cao shows up and defeat him instead.
4 - Cheng Du
5 - Yi Ling