I'm playing as a bard that has a proficiency bonus that's added to Dexterity. When calculating the character's Armor Class, do I add only the base Dexterity modifier or do I add the base Dexterity modifier and the proficiency modifier?
"a proficiency bonus that's added to dex."
That's not a thing. Your proficiency bonus might be added to some dexterity checks if you're proficient in a dex-based skill, and it is to your dexterity saves. But there's nothing like what you describe, where proficiency's just added to everything that references dexterity. (Even Jack of All Trades only applies to ability checks.)
Now, to calculating AC: the armor you're wearing or the spell you're using or the race or class feature implicated will tell you how to calculate your AC. If you would have multiple ways of calculating AC they function independently, and you choose one. Assuming you're wearing armor you might add some or all or none of your dexterity modifier as part of the AC calculation. Nothing I know of would bring your proficiency bonus to the AC-calculation table.