Alchemists' Toybox- Gunpowder; Class Options (D&D 5e)

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Fighter

Add Fighting Style option;

Overload

You gain a +2 to damage you deal with firearms.

Rogue

Add to starting weapon proficiency; heavy and light pistols.

Rogue Archetype; Gunslinger

Tools of the Trade

At 3rd level you gain a light or heavy pistol (your choice). Work with your DM to determine how it came into your possession; for example it could be a family heirloom, a discovery from an exotic location, or the results of your own tinkering and crafting.

Serpentine Smith

At 3rd level you are proficient with alchemist’s supplies. Further, after a short or long rest you can craft a number of cartridges from materials you have and find. After a short rest you can make a number of cartridges equal to your proficiency bonus. After a long rest you can make three times your proficiency bonus. Unused cartridges expire after your next long rest, or 24 hours if that comes first.

Iron Gaze

You are trained in Intimidate. As an action you can force a target to make a Charisma save against a DC of 12 + your Charisma modifier + your proficiency bonus. If the target fails this save they are frightened of you for one minute. If they are successful on this save they are immune to your use of this ability for 24 hours. At 13th level success does not make your targets immune to this ability, and at 17th level you can force targets that fail their save to drop anything they are holding and only use the Dash action to move as far from you as possible on each of their turns unless they have nowhere they can move.

Swift Shot

At 13^th level, you can use your bonus action from the Cunning Action rogue ability to draw, fire, and re-sheath a light or heavy pistol. You suffer a -2 to the attack when using this ability with a heavy pistol.

Always Ready

At 17th level you ignore the loading quality on firearms.

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While full classes are overly focused, there are several ways to play a character that emphasizes the use of gunpowder and firearms. I've already presented several feats, just for this purpose. Even the use of a background might work, though I've yet to see a firearm-related background that isn't character-narrowingly obsessive about that single trait to the point of feeling like it should preclude adventuring. Meanwhile, these are a few possible tweaks to help integrate firearms use.
The archer fighting style is an outstanding choice for a firearm focused fighter, and using a Champion archetype fighter who focuses on ranged weapons adds some complexity and options into what is otherwise a very stripped down archetype. I specifically created the Overload fighting style to fill the second fighting style slot that such fighters get.
I suggest house ruling that all rogues get proficiency with pistols. For me, it's a logical extension of their proficiency with hand crossbows.
Then we have the rogue archetype. While I think full classes are too focused, archetypes in all their forms are about exactly that sort of focus. Rather than trading general (play)ability for being related to a specific type of equipment or action, they provide bonuses to that particular style and are balanced against each other based on how they shape the larger classes' approach to its niche, rather than having to find ways to fill an entire niche- which "guns" are generally too narrow of tools to do. What I've mostly seen are would-be sorcerers skinned as rogues. Here, I've simply left the base character as the (generally useful) rogue, and made their special skills related to the use of (and imagery associated with) firearms.