If you want to drive with a specific livery you have to sign up to either a season or open event and also participate in that event. Outside of events the server will give you a random skin.
This includes the looping practice in the lead up to an event - when you join the closed event practice you should get your assigned livery.
When registering for a season there will be an option to upload a custom livery. Once a livery has been uploaded, it will remain for you to use again in future races (and will be part of the pool of random skins in open events)