Your low-Dexterity character is a klutz. You'll want to demonstrate this in minor ways at first, to let your party get used to the idea.
Describe how your character is admiring apples at the fruit cart in the town market... and then drops one into the street, where it gets crushed under the wheel of a passing merchant's wheelbarrow. The fruitseller, unimpressed, tells your party, "That'll be a copper." No big deal for adventurers, sure. Do this a couple times. Maybe you accidentally knock a small animal cage open and have to catch the animal, pay for it, or prevent it from causing further market mayhem. Maybe you drop a dagger at the blacksmith's... luckily, no harm done, no fingers lost... but the blacksmith is now annoyedly huffing at your party; try haggling after that.
Then, after a couple successful adventures, offer to go potion-shopping... and see if your party will let you go alone. Say that somehow, they let you go. Roleplay being drawn to the bottle of a potion on display. Mention that you want to pick up the bottle to examine the craftsmanship of the glass, and how it refracts the light... and see how fast your companions interject with a "Don't touch that!"
Let's not talk about that one time, in the dungeon, when you were near the cauldron of boiling acid.
Or, maybe your character knows he or she is a klutz. You offer to sit in the cart instead of taking a turn clearing the trail ahead. You stand back when it comes to the heavy lifting. "Oh, it's best for all involved if I don't," you say, when the fighter wants you to open the door so they can charge in, or to the rogue who wants you to hold onto something so they can disarm the trap. You get a reputation for being lazy or too fussy, but really you know what'll happen if you lend a hand...
Did your character grow up rich? Maybe you are fussy. Maybe you've had others to do the manual labor for you all your life. You'd rather your party members do things for you, too, because old habits are hard to break, but you realize that it's not their job. But you're becoming really good at making excuses and being really polite, finding new ways to phrase your requests every time it comes up. Your party is going to catch on eventually, so expect to soon be on your own with these things, and like it.
Maybe your character overcompensates. At festivals, you makes a beeline to knife-throwing games because all you need is a little more practice, obviously. You never win anything, of course, unless you get terribly lucky. Then you let everybody know of your exceptional deed! This inflated self-recognition happens in combat, too. You roll a natural 20 (or whatever the equivalent is in the system you're playing) and suddenly your character is commenting on what a great strike that was, did you see that? You keep going on about it for a round or two, until your party reminds you that you're still in combat and can it wait, maybe, until everyone is out of danger? Okay? Please?