First thoughts:
Overview
Saving throws: Should have one of common Dex, Wis, or Con, and one of uncommon Int, Str, or Cha. You have Dex and Wis, both common saves.
Weapons: You start with a rapier or shortswords, but aren't proficient in them until level 3. You should start with prof in them, and let the subclass optimize the player's choice. Also, for a marksman, not having access to other thrown weapons seem off (handaxes, javelins, etc).
Bullseye: Normally saving DCs are based on the primary spellcasting ability, but there are exceptions, such as Pounce (and being knocked prone) as a Strength saving throw. As you are throwing, shooting, or being Precise, maybe change this from Wisdom to Dex. This may cause this class to be completely SAD in Dex though unless they are a Bone Reader...
Deny the Shot: A mix of Dodge (action) and Monk's Deflect Missile (reaction). It may be a bit too situational to use instead of Dodge. +4 AC is very similar to disadvantage on attacks, which could be easier to word.
Eagle Eye: As a Surgeon it may be more Thematic to give Advantage on Investigation over Perception, but that's just me.
Bloodletter: Creatures with Int 12 are not always common in every setting. Even base humanoids have an average of 10. Taking an action to stop the bleed damage could be enough, stopping creatures from doing so or requiring a check may run into problems (shoot a few guys then run off while they bleed).
Blowgun Master: Increasing the damage by one die at this point doesn't do much. Seems like you are trying to shoehorn the blowgun into the class, and a d6 isn't going to do it.
Lethargic Perforation/Vivacious Touch 11th Level: I like the idea of using Pressure Points as a class feature. But level 11 is a huge milestone for most classes and increased damage. Precisionists only get to use Haste (I don't see Slow being used over it much) once a day.
Surgeon:
Quick Stitch Up: This should be Medicine skill and Healer's kit
Hemorrhage/Dueling: Give Dueling at lvl 3. Hemorrhage seems like a good bump to damage.
Hindering Incision: I like the theme of this. If a target uses Dex for attack, maybe they halve their damage for that as well.
Blood transfusion: Now I just think of Dexter
Swift Blade:
Prof: again, give Shortswords as class prof.
Crippling Prick: 1/2 speed may be easier to remember, but I like the theme of this.
Two-Weapon Fighting: Maybe at 3rd level? Seems to get it late.
Fall From Grace: Pretty powerful to reduce flying speed to 0. Maybe now when you use Crippling Prick all speed (flying, etc) is reduced by half, and walking speed to 0.
Speed Stealer: Seems to change the class suddenly from being precise and quick to mystical (again, this is just me). Also may run into issues if you are flying around and your party wants to kill the dragon, like right now!
Somewhere in this subclass there might be room for a basic walking speed bump for 5-10ft? Maybe change theme from Swift Stroke to Pressure Points (with Chi and flow and etc, but that's pure fluff)
Bone Reader:
Spellcasting: 1/3 spellcasting fits well, and I like how this class is available. But is the Wizard spell list (aka the complete spell list) the best choice? Studying from bones gives me an image of memories and lives lived, not tomes of knowledge. Instinctual learning, like the Warlock, Sorcerer, or Druid list (even Cleric) seem a little more thematic (again, I like themes, and this is just me)
Study the Bones: The time, material limit (100 year old bones are always around) and the limited vision seem weak, even for a ribbon feature. Talking to the spirit may help. Also, the other subclasses get significant damage increase at this level. Regaining spells is useful, but with only two level 2 spells at lvl 7, and the need to have 100 year old bones to get two slots back, it will take a while for this feature to be helpful.
The Third Eye: If you have Darkvision maybe make it Superior (to 120 ft)
Festering Wound: Other than getting past non-magical/necrotic resistances, this doesn't do all that much until lvl 17. The Precisionist could gain necrotic resistance himself, immune to disease, etc as well.
Wither the Soul: Imposing vulnerability to one target for your attacks (and not all of your spells) for 4-5 rounds may work against a BBEG, but isn't great for a level 17 ability.
One other thing about the Bone Reader is the player will need to keep track of both his spellcasting and his Bullseye Charges, leading to lots of bookkeeping.
Thoughts:
Overall I really enjoy the theme, and everything listed is my own brief thoughts looking over it. The best way to balance it is to look at other classes, but I am AFB. The class seems built around DEX, with Wisdom being available for some abilities or Bone Reader. There is no support for using Strength (or thrown weapons) and there are already several classes and builds that utilize DEX/WIS combos in 5e. Replacing Wisdom with Int may separate this class from looking just like a specialized Monk, Ranger, or Rogue. But I think you are able to give enough depth and identity (especially within the subclasses) to have a good class on it's own, so good job!