Council requests will appear at random times via the situation room. These will involve a council nation requesting either alien technology/artifacts, items you can/have engineered, or satellites to be launched in exchange for rewards. Each request will carry a certain time limit allowing you to build/obtain the required items. Once you meet the requirements, the request will be completed and the achievement will unlock.
TIP: While most items will be fairly easy to build/obtain within the time limits, satellites will take longer to build then the allowed time, so it’s a good idea to always have a satellite in reserve so it can be launched quickly.
Also, the council requests are random, so if you receive a request that you don’t believe you can fulfill, don’t go out of your way, instead wait until an easier request comes along.