***General Challenge Mode Info: The achievement description tells you everything you need to know about how to get it... simply beat the record for each challenge. The characters in challenge mode have set levels, and set skills. You do not need to level any of your characters up to do better at challenge mode. You must also unlock any characters you wish to use in Challenge mode.***
The fastest path I can see, and because I got the achievement for completing it this way I didn't see a reason to refine it, is to run North from the start to the first base you come to. Go immediately west to the next base, then south after that. Continue southeast until you get to the base in the southeast corner then go north on the east side of the river to the last base in the center north of the map. The path you take would look like a small outward spiral if you were to draw it on the map.
Good fast characters are Gan Ning and Zhang He, though there are others with decent speed. Zhang He is my favorite for this, because his running Y attack doesn't slow him down much. Ever 10 soldiers you kill gets you a 10 second speed burst. It is *NOT* worth stopping to kill soldiers as you go, just kill some as you run by. Bannermen will drop 10 second speed bursts each kill, but are not worth going out of your way for. Siege weapons will drop 30 second speed bursts, and are worth stopping for, but not worth going way out of your way for.