Design ship.
Lock the design. - Not technically necessary, as the design is locked automatically when a shipyard is tooled to the design or the first ship is placed under construction, but helps prevent accidental alterations.
On the manage shipyards tab of the colony screen, select the shipyard from the list. - Ensure the shipyard is large enough for your design and of the correct type, commercial yards cannot be tooled to a military design.
From the Task Type drop-down, select "Retool for Selected Class" and hit the set activity button. - If this is the first design to be tooled to the shipyard it will be instantly completed. If changing from a previous design it will take time, money and resources to change to the new design.
You may need to hit the refresh all button or close and reopen the colony window at this point, as I've had occasions of being unable to select the newly tooled design immediately.
Select the ship from the New Class drop-down. - You will see listed on the right the minerals that will be required to build the ship. You do not need to have these currently available to start the construction but it will stall if one of them runs out until new minerals are mined or arrive.
Select a task group that is at the same location as the shipyard, preferably one that you will use just for new ships, transferring them to another task group as they are completed, as having an under-construction ship assigned to a task group prevents it from carrying out many orders.
Hit Default Fleet to save the task group assignment to the shipyard.
Hit the Add Task button. - At this point if you have any stockpiles of components produced by your industry that are used in the design, up to the number used in the design will be removed from the stockpile and credited towards the ships completion, reducing the remaining cost and hence time proportionally. Components produced after a ship is under construction will not be used to further complete it but will remain until the next ship begins construction.
Each shipyard can have a number of ships equal to it's slipways under construction at the same time, any further ships ordered will be added to a queue and construction commenced as soon as a slipway becomes free. Shipyard modifications such as increasing capacity, adding slipways or even retooling to a new class design can be carried out while ships are under construction without effecting them. If retooling, new ships can be started using the current design right up until the retooling finishes.