Once you have completed a Campaign mission for the first time, you will have the option to use a different character skin on that mission in future.
The option for this is at the bottom-right of the Loadout menu before you start a mission. The normal Player Loadout screen on the Main Menu doesn't have this option, as it is dependent on you having finished the mission at least once.
Please note that not all the characters available in Multiplayer modes can be chosen for Campaign missions.