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?
So here's my issue: I like my character, I like our party, and I don't want to pull a 180 on my character and make him nice or throw away important motivations for him.
Well, it sounds like your character just may be evil, or at least on the evil side of neutral.
That doesn't mean he has to do evil things, especially if he has a reason not to.
And, if he wants to stay with the party, he probably does. You've already had an in-character conflict where one character "stormed out" and another "left the room disgusted." That ought to be a pretty good clue to your character that, if he wants to hang around with these guys, he'd better start to act nice, even if he's only doing it to keep the other party members cooperative.
Even if your character was a complete psychopath who loved kicking puppies, if he was sufficiently smart he should be able to realize that there are situations where it's better to heroically save the puppies instead. And it doesn't sound like your character is anywhere near that bad.
Basically, you're playing a conflicted character. This can be a lot of fun, if that's the kind of thing you like.
It doesn't matter that the cause of the conflict may be (at least initially) external; even so, it's a source of mental conflict for your character. On one hand, your character worships an evil god. Even if his choice is fundamentally based on pragmatic reasons (power!), a pact with Cthulhu will surely have some influence on him, tempting him to more evil acts and means (not that a hunger for power couldn't do that all by itself). On the other hand, he's also fighting to save the world (even if it might be for his own ends), and has teamed up with a bunch of more noble, good and squeamish types to do so. This means that, whatever means he might want to employ in the pursuit of his goals, he now also has to consider their effect on his fellow party members.
And, of course, once he starts to consider the way his actions are perceived by others, he might also gradually come to realize that there's also a side to himself that doesn't like what he's doing. Maybe not instantly, but after a while. Morality has a funny way of growing on you like that — from "I have to be nice or I'll get punished" to "hey, other people are actually nicer to me if I don't act like an asshole" to "you know, I really should be nice just on principle, because it makes the world a better place."
None of this means "pulling a 180" on your character, or at least, not in a way that he wouldn't have good in-character reasons for. Sure, the conflict with the other party members (and possibly subsequent reflection) might be the trigger that makes your character realize that he needs to drastically change his behavior in order to achieve his goals, but his motivations will still be the same as before.
Best Answer
Having at least one low stat is pretty much a given for any new character. Depending on if you roll for your stats, or if you "buy" them using the Ability Score Points Cost Table (PHB p.13), and a little bit of balancing using racial modifiers, you might be able to get away with your lowest stat being 10, but sometimes you have to live with an 8 or a 9.
Each Ability Score is important in it's own way, and depending on your class, you will have a Primary Stat, (Eg, a Barbarian might focus on Str, a Bard would focus on Cha, etc.), and you can see which Ability Score is important for which class (PHB p.12). In relation to Con (emphasis mine):
Most races (but not all) do get a natural bonus to this as well, so if you do get a negative Con score, you can balance it out with these bonuses.
Now, to answer the question: What negatives can come from a negative Con Stat?
Short answer: Everything. DO NOT MAKE CON A DUMP STAT
The longer answer: Everything. Constitution is known mainly as the "HP" stat, as every level, you increase your Max HP by rolling (or choosing the average) increase, and adding your Con Modifier. This means that a negative Constitution can actually decrease your health, depending on just how bad your Con stat is. However, assuming it's only -1, let's do a quick sum:
On top of this, all of your Con checks, and saves are also at a negative. The PHB also lists the sorts of checks that Constitution are responsible for (PHB p.177):
In the case of holding your breath, this means that you could actually die quicker while drowning, or suffocating (PHB p.183):
This means that with a negative Con, you can only survive 30 seconds, then drop to zero HP after 1 round.
In other cases, such as marching for extended periods of time, or going without rest, if your DM is playing with the Exhaustion rules (PHB p.291), these can stack up fast.
More specifically, for a Cleric, Constitution can also affect your spellcasting abilities, specifically with spells that require Concentration; as they need to be cast, and held for longer periods of time (PHB p.203):
In a situation where you are trying to hold a spell that requires concentration, taking a hit can impede this, requiring you to make a Constitution saving throw, in order to maintain concentration:
This means that with a negative Con score (-1), the minimum automatically raises to 11 on the dice, to pass a check, provided you take less than 20 damage from a hit (in which case you'd like be incapacitated, due to a low health score anyway).
So in summary, DO NOT MAKE CON A DUMP STAT.