The Artificer
I agree with divibisan that a Rogue may be your best option, but I'll add an alternative. The Artificer (from Unearthed Arcana: Artificer) does not get extra attack, and much like the Rogue, focuses on creating a high damage output on a single shot per round. A light re-skin of the Gunsmith Artificer Specialty from the Artificer class in Unearthed Arcana may fit your needs. The specialty utilizes a Thunder Cannon (Firearm) that could be re-skinned as a crossbow (See below, emphasis mine).
Thunder Cannon
At 1st level, you forge a Thunder Cannon. It is a ferocious weapon that fires leaden bullets.
You are proficient with the Thunder Cannon. The firearm is a two-handed ranged weapon that deals 2d6 piercing damage. Its normal range is 150 feet, and its maximum range is 500 feet. Once fired, it must be reloaded as a bonus action. If you lose your Thunder Cannon, you can create a new one over the course of three days of work (eight hours each day) by expending 100 gp worth of metal and other raw materials.
The ability initially closely matches the heavy crossbow martial weapon in range (150/500 vs 100/400) and damage (2d6 vs 1d10, both piercing), but rather than having the loading property, it requires a bonus action to reload.
The class abilities Thunder Monger, Blast Wave, Piercing Round, and Explosive Round could all be flavored as specialty crossbow bolts. The Thunder Monger ability, which can be used for every attack, roughly matches the rogue's sneak attack damage.
You can read more about the class here. You should ask your DM if Unearthed Arcana material is allowed in their game and if they approve of your changes.