I was a little caught off-guard during the most recent 5e session in a campaign I run, so I was hoping to get some input here. One of the players snuck up on a villainous NPC and stabbed them in the neck with the intent of killing/making them bleed out. Of course, the roll was a natural 20, so it succeeded. Not wanting to lose this NPC over a dagger's d4 worth of damage, though, I had them start bleeding. They (the NPC) had to make a Constitution save: 2d6 damage on a fail, half on a success. I'm curious, though, if there are actual rules on these kinds of situations? Would the NPC just have died outright? Are there bleeding mechanics?
There's a few points here. First off, even on a critical hit, it's up to DM fiat as to whether the NPC just dies outright, or takes HP damage, or whatever. If this is an important, named NPC, you probably want to have them take damage rather than just declaring them dead or dying.
If you decide to have the attack deal critical damage against HP, then do that. The player rolls all the damage dice twice and adds it up, and that's the damage dealt. Yeah, that might mean a sneaky attack from behind deals a whopping 2d4+3 damage, which is eminently survivable. (Unless the PC is a rogue, in which case it might be many d6's in addition...) If your NPC doesn't drop to zero hit points, there's not even a question -- they aren't dying. Depending on the damage dealt and the NPC's hit dice, they may not even be seriously hurt! It would be up to you as DM to describe how they sense the attack at the last moment and twist so it hits their shoulder, or something like that.
Second, yes there is a sort of 'bleeding out' mechanic built into the game; it's called a Death Saving Throw, the same as for a player who falls to zero hit points. Roll your d20 until you get three successes or three failures.
But third, if you're the DM, you decide if NPCs live or die, not the mechanics. Suppose mechanics are off the table, maybe you didn't have any stats or HP for the guy in question -- you should be the one deciding if you want to let the PC kill the NPC or have him somehow survive (and if so, how). If he shows up later with a huge scar across his neck, you don't actually have to explain what die rolls he made to do that. It just is. The game should be 'fair', but you aren't required to account for and justify every decision you make behind the scenes to advance the story.
There are no default rules on causing bleeding wounds, but that doesn't mean you can't steal some if a situation comes up that warrants them. Remember to go with what makes a cool and fun story, since that's (probably?) what you're playing for.
So for a next time, if you want to cause wounds that bleed profusely, consider stealing them from the Bearded Devil:
Glaive: Melee Weapon Attack: +5 to hit, reach 10 ft., one target. Hit: 8 (1d10 + 3) slashing damage. If the target is a creature other than an Undead or a construct, it must succeed on a DC 12 Constitution saving throw or lose 5 (1d10) Hit Points at the start of each of its turns due to an Infernal wound. Each time the devil hits the wounded target with this Attack, the damage dealt by the wound increases by 5 (1d10). Any creature can take an action to stanch the wound with a successful DC 12 Wisdom (Medicine) check. The wound also closes if the target receives magical Healing.
or the Sword of Wounding:
Once per turn, when you hit a creature with an Attack using this Magic Weapon, you can wound the target. At the start of each of the wounded creature’s turns, it takes 1d4 necrotic damage for each time you’ve wounded it, and it can then make a DC 15 Constitution saving throw, ending the effect of all such wounds on itself on a success. Alternatively, the wounded creature, or a creature within 5 feet of it, can use an action to make a DC 15 Wisdom (Medicine) check, ending the effect of such wounds on it on a success.
The Sword of Wounding is a Rare magical item, so that might also give you an idea of how valuable an effect like this would be. Still; sometimes the situation calls for a major bad guy clutching his neck while blood sprays everywhere, and when it does, the above should help you.