The first option to pick a government in the standard game is when unlocking the 'Political Philosophy' perk in the Civics Tree.
You can progress towards this faster by increasing your culture per turn as that is what helps towards your policy improvements. So any improvements or buildings that increase culture (the purple music icon) will help towards this.
Such as monuments, which are some of the beginning buildings in the game to help getting this quicker.
After unlocking it, you have the option for 3 governments and picking any three will net the achievement.
The streamline to get here in the tree is: Code of Laws >Foreign Trade / Craftsmanship > State Workforce / Early Empire > Political Philosophy