Stellaris

There are three options here:

a) defeat empires in war, and forcibly turn them into vassals
b) become so powerful that empires voluntarily become your vassal (this is requested through the diplomacy menu)
c) enlighten primitive civilisations, they automatically become your protectorate and then (a few decades later) your vassal.

The easiest method is (c). Create a medium-large galaxy with no AI empires. Set primitive civilisations to 5x, and habitable worlds to 0.25x, then get exploring and colonising. Once you find a primitive civilisation (in a 600 star galaxy you are looking at approx. 15 across the galaxy), build an observation post and set it to technological enlightenment. Once they are enlightened, the empire is automatically your protectorate. After they discover 50% of your research technology (this will take approx. 50 years) they become your vassal. You can be conducting this process for three civilisations simultaneously, to get the three vassals needed for the achievement.

Note: technological enlightenment is not possible for fanatic xenophobe empires (both by them, and for them).

I saved this one for later thinking it would be a pain but it is actually really easy. Expand out and colonize three planets at least 5 (maybe 6) jumps away from your home world in all directions. When they become colonies they should show up in the Frontier Section of your asset list on the right. Go into the planet and next to the sector name is a create sector button. Click it and go through the menu to make a sector. Once you have three extra sectors past your starting core one, go to planet and sector management on the left menu and click Create Vassal on all three.