It's easy to notice the Barbarian who can break down a door with their bare hands, the Cleric with the wisdom rivaling the gods, the Bard who can charm your pants off, the Wizard who has memorized all but the last two digits of Pi, the Druid who can tank a hit, and the Rogue who—
Wait, what Rogue?
High stats are easy to play, as they literally play to your strengths. But it's fun to also play to your weaknesses. When I hear of people playing their weaknesses, though, they tend to stereotypical, or entirely too weak.
I'm going to start by saying: NEVER MAKE CONSTITUTION YOUR DUMP STAT. I don't know why you would ever do this, but this is a bad idea. However, for the two people who need this for some character backstory, there are ways to play this in character.
A character with a weak Constitution would be frail and almost sickly. I picture an Elf who is stereotypically thin, but a bit moreso. Otherwise, they'd perhaps have a persistant cough or ailment that's not enough to hinder them but if they get hit, it's obvious how often they "fall down" (because they're at 0HP so quickly).
This is another stat that I personally feel should never be your dump stat. Where Constitution can hinder HP, Dexterity can hinder your AC — as well as three (albeit situational) abilities, and potentially your aim and damage. But, again, if you must...
This character is clumsy, perhaps. Not enough to trip and break a bone. (unless your Constitution is also low. What did you do, roll?!) I would also probably play this character with a slow speed (flavor, not mechanics). Maybe paired with a high Wisdom, they are trying to be overly cautious. Their aim is also horrible. Play into it: maybe they like to throw rocks, but they're always breaking windows accidentally.
This is historically my favorite dump stat. This is probably your caster's weakest stat because they just don't need it; you're focused on getting your mental stat to 18 and since you're not even going to wield a dagger, theres no need for Strength. (Sidenote: I've been playing more with making this my next weakest stat, and choosing another mental stat as my weakest, depending on the flavor. Not every caster needs to be "weak." I've grown more accustomed to setting this to 10.)
This is easy to roleplay. They are the ones who can't help break down the door or lift the fallen boulder or carry a comrade.
Where Strength is a caster's preferred dump stat, unless you're a Warlock or Sorcerer, this is your preferred dump stat. Fighters, Druids, Rangers, even Clerics. What need for speaking do you need to be when your party has a Bard? You just want to hit things.
Characters with Charisma as their dump stat stereotypically are shy people. But there's more than one way to play this. Maybe they do talk, but they trip over their words. Maybe they're from a far off land and their communication style just doesn't mesh well with the local populace. My Razorwitch (a reflavored Eldritch Knight) has Charismas as his dump stat. And I have him talk all the time. He's a chaotic little Halfling and is eager to do everything, even if he messes up. It's also popular to have Charisma represent looks, so you can also have your character be unimpressive, bland, nondescript, easily hidden in a crowd.
Only two classes need Intelligence: Artificer and Wizard (with Eldritch Knight and Arcane Trickster thrown in for funsies). I consider this a "safe" dump stat. There's very, very few spells that requires an Intelligence saving throw, and if your character is not actively looking for something (Investigation) or knowledgeable, dump away.
Characters with a lower Intelligence don't drool when they talk (that's probably more of a low Charisma, honestly), and they don't have trouble doing simple math. In the Point Buy system, an 8 does not mean your character is dumb as a pile of bricks. Maybe they just don't have formal training, or certain concepts just escape them. You can honestly play off a low Intelligence very nonchalantly. You may never even need to bring it up.
I (oddly?) name this one last, but that's only because of the (over) importance of a single skill: Perception. Even having a +0 bonus can feel painful when you fail a perception check. Having that bonus as a negative will be unbearable after a while. I will consider making this a dump stat, but would probably choose Charisma, Intelligence, or Strength instead.
But if you must, characters with a low Wisdom may be easily distracted. Maybe they daydream often. As the Perception skill will tell you, maybe you just have a hard time seeing something off in the distance. Maybe you're quick to trust others despite obvious red flags.
It's never too late to start adding a bit more flavor to your roleplaying, or you can start fresh with your next character. Whatever you do, embrace the weaknesses and have fun!