Space civilizations demand bribes when your relationship with them is below 0 points. This is easier to do with Warrior and Zealot Empires, since they start with -30 (or -15) with you. This achievement will cost you many Sporebucks, so before trying to get this one, make sure you have a lot of Sporebucks with you, because you can't cheat.
Gifts and peace offerings count as bribes.
This can be done across any number of games.
Just get a lot of money and keep giving it as 10,000 gifts to empires you meet. This is also useful when you are a diplomat as it helps you make alliances.