Every background provides you with clothes appropriate for your character. Because it's listed as "equipment" and because the Adventuring Gear table provides weights for several types of clothing [PHB pg. 150], it's reasonable to conclude that this mass must be accounted for when ensuring you remain under your carrying capacity, which is 15 lbs times your strength score [PHB pg. 176]. That's simple enough, and the 15 x STR limit is high enough to be effectively irrelevant unless you're trying to haul away a dragon's hoard.
However, if you're playing with the encumbrance variant [also PHB pg. 176], things can get more problematic. In that scenario, there's a movement penalty imposed for carrying more than 5 x STR, and quite severe penalties for carrying more than 10 x STR. For this reason, I prefer my characters to regularly carry no more than 10 x STR, and then to configure their gear in such a way that they can drop their pack to leave themselves with a "combat load" (of weapons, armor, precious items, etc.) that's less than 5 x STR.
However, I find myself pushing the limits when I have a character with average-to-low strength (say 8 or 10) and medium armor (usually 20-45 lbs). In this scenario, every pound matters, so I need to know: Is the clothing in a character's equipment actually a change of clothes, able to be stored in their pack and dropped during combat? Or is it added mass that they're wearing (under or over their armor) into battle?
Naked adventurers need not apply. (Bards should put their clothes back on. And barbarians . . . well, you do you.)
After further consideration, I've noted that the various sets of clothing in the Adventuring Gear table weigh from 3 to 6 pounds. I believe this is significantly more than a single day's set of modern clothing weighs, which would support the idea of this being extra clothing, but I have no actual data on the typical mass of modern or medieval clothing to back this up.