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):
Measures: Health, stamina, vital force
Important for: Everyone
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:
- At level 1, a Cleric starts off with 8 HP + Con Modifier. 8-1=7 Max HP. Not a great start.
- If you choose to roll for your HP increase at each level, (a Cleric uses a d8) you can get anywhere between 0-7. That's right, you can increase your HP by zero, if you roll a 1 on the dice.
- Alternatively, you can buy the average, leaving you a 4. (Average is 5 + Con)
- By the time you reach Level 4, this means that you can have anywhere between 15 HP, and as low as 7 HP, when you can then choose to increase your Stat, if you're not already dead (A CR4 creature is expected to do 27-32hp of damage per round - against someone with 7 to 15hp they may be dead before they can act - courtesy of Dale M).
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):
Constitution measures health, stamina, and vital force.
Constitution checks are uncommon, and no skills apply to Constitution checks, because the endurance this ability represents is largely passive rather than involving a specific effort on the part of a character or monster. A Constitution check can model your attempt to push beyond normal limits, however.
The DM might call for a Constitution check when you try to accomplish tasks like the following:
- Hold your breath
- March or labor for hours without rest
- Go without sleep
- Survive without food or water
- Quaff an entire stein of ale in one go
In the case of holding your breath, this means that you could actually die quicker while drowning, or suffocating (PHB p.183):
A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).
When a creature runs out of breath, it can survive for a number of rounds equal to its Constitution modifier (minimum 1 round). At the start of its next turn, it drops to 0 hit points and is dying.
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):
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
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:
Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
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.