A Capital City is an extra large city that has no dome over it. In order to unlock it you need to Terraform Mars sufficiently in order to have a breathable atmosphere. Breathable Atmosphere occurs at 95% Atmosphere and 50% Temperature.
Temperature is improved by using GHG Factory and Core Heat Convectors.
Atmosphere is improved by burning Waste Rock in a Carbonate Processor, having Magnetic Field Generators also slows atmosphere loss.
Special Projects can also be used for improvements.
Once you are at 95% Atmosphere and 50% Temperature you will be able to build the Capital City. Once it has finished building the achievement will unlock.
More detailed Terraforming info can be found on the wiki located here.
https://survivingmars.paradoxwikis.com/Terraforming