The best characters to use are prepromotes, units with high movement, and fliers. Seth will be your main fighter for the first part of the game. Flying units with manipulated stats are extremely powerful, so train at least one of them. He/she should be promoted to wyvern knight to abuse the Pierce skill.
RNG manipulation is especially important in a TAS. The RNG state is stored at 03000000-03000005, and consists of the last 3 (16-bit) random numbers that were generated. The formula for the RNG is a simple bitwise XOR of those RNs shifted in a certain way. You can waste lots of RNs by drawing paths, and longer paths waste more RNs. Use this to manipulate stats, criticals, dodging, Pierce, and anything else you can think of.
Finally, there are some runs you might want to check for reference:
My TAS of Fire Emblem (Rekka no Ken):
thread,
full test run,
latest WIP (chapter 10) - sort of abandoned, but I plan to continue it eventually
cheetah7071's abandoned TAS of Sacred Stones:
thread,
latest WIP (chapter 13)
Molotov's speedrun of Sacred Stones:
http://speeddemosarchive.com/FireEmblemTSS.html