Simply last 8 years and 4 months as governor without declaring independence, running out of mandate, or otherwise losing.
There's plenty of opportunities to extend your mandate, just keep an eye out for challenges appearing above your palace and accept mandate extensions as your reward for completing them.
Going into the journal page for your budget is a good way of tracking how far along you are. The numbers along the bottom of the graph represent each year of your mandate. The achievement should pop once you're 1/3rd of the way into your 9th year.