Buildings can be built by speaking to a town’s architect but before you can build a building you need to purchase a licence from the architect in the town you wish to begin building. To do this you need to be the rank of sailor, and meet the reputation requirements (both in that town, and with the host nation). The required reputation increases each time you purchase a license in a town. Once you meet the requirements you can purchase a license that allows you to build warehouses and houses.
From there, once you build a combination of 3 warehouses or houses, you can purchase a second licence that allows you to build businesses. Each building will require a supply of wood and bricks taken from the town’s dock, so it’s a good idea to set at least one trade convoy to ‘collect building materials’. In order to reach 250 total buildings you will have to buy licences in several towns. This achievement will come naturally as you progress through the game and will carry over between saves.
TIP: To speed things up, purchase a licence in a town that produces Adobe Bricks, (Florida Keys is particularly useful as it’s centrally located) and create several of your own brickworks to ensure a steady supply of bricks, as this will be the main resource that will halt construction.