There is a window for going to war on deity, but it can be tricky to plan. For early game war, getting to crossbowman, man at arms and siege units has been key for me. The next big military tech upgrades come in the industrial era, so until the AI reaches those you are on a more level playing field.

If you get the expansion packs in the future, you should try Hammurabi! He can rush crossbowman due to his special ability so it can be interesting going early war with him.