You add them because they are included in your character level, but you don't double-count them (unless you use both abilities at once).
The spell specifies character level, which is the sum of all your class levels and hit dice (source). Thus, a multiclass Paladin/Cleric would add up their Paladin levels, their Cleric levels, and any other levels they had, and add that number to their damage when using the Smite Evil ability granted by Visage of the Deity.
However, this has nothing to do with the fact that Paladins can naturally Smite Evil - it's a consequence of what "character level" means, and would work just as well for a Fighter/Cleric. When a Paladin/Cleric casts Visage of the Deity, they effectively have two different Smite Evil abilities - one (from Paladin) that adds their Paladin class level as damage, and another (from the spell) that adds their character level.
An additonal note, per the comment thread below: Because Smite Evil doesn't take its own action to activate, a Paladin/Cleric probably could use both of these Smite Evil abilities on the same attack. In that case, you would double-count the Paladin levels, because you'd be adding your character level for the spell-granted version, and then adding your Paladin level for the Paladin version.
The bonuses to attack might or might not stack in this case - it depends on how your DM feels about the "same source" stacking restriction (if their source is the ability that grants them, they're coming from different sources; if their source is "your Charisma bonus," they're the same source).
Ask your DM before you try this at home.