The initial design and visuals:
Looks like a cross of Graves and Twisted fate - the cowboy thing is mostly covered so it could be rough to try to get that "I have this theme but I'm still different from these other guys" thing working.
His lore:
I know it's not done yet, but it feels kinda similar to the situation of Rengar and Kha'Zix, even to the point of him losing his eye.
On to the abilities:
passive: mmm this passive just doesn't seem worth it; I mean, you'd be better off playing Draven since he continually gets his passive bonus to damage (granted it's not on Q but still) as long as he can catching axes. Basically in a game where ranged champs have unlimited ammo, you have to get huge bonuses to damage to justify not being able to attack for 2 seconds.
Q - So, it's a longer range on-click for some amount of mana. Like Gangplank Q? Could work, I suppose, but a little boring perhaps. Also taking the rifle off his back to shoot is soooo vayne.
W - This is an ability which is supposed to make up for the huge-penalty-for-not-much-benefit passive, but then leads to you having to reload again faster. Not good on this kit in the slightest.
E - Vayne's tumble exactly. Not good not good. We have 4 types of "dashes" on our marksman champions -
I dash a small distance and gain a damage bonus on the next hit. Her ability is not considered "spent" until she attacks with it, which reinforces her aggressive chasey playstyle.
I dash a small distance and gain attack speed for a few seconds. This is mainly used to set up for his shotgun blast, while also giving him follow up.
I blink a respectable distance and do magic damage in an AOE around me. Fitting into his mage-like theme, Ezreal has an actual teleport and gains magic damage from using it.
I dash a moderate distance, and have reduced cooldown on my dash for auto attacking. Lucian's cooldown
All of these marksmen have dashes, yes, but they all fit the character and they're all different in some way. Yours... kinda doesn't.
R - In a nutshell, this is "Reload his gun, his attack speed becomes 2.0 for the next 6/8/10 attacks in which he does x3 damage." This is a crazy powerful steroid while also being somewhat boring in execution. Also I thought he just lost his eye, why is it "cursed" now?
Overall thoughts:
This is going to sound a bit harsh but right now your kit just feels very uninspired - I can see that you want to play around this 6 shooter playstyle, but you've taken a very punitive approach to get there.
I think your core idea is not necessarily a bad one, but the 6 shot mechanics should be more rewarding and less penalizing. Just off the top of my head suggestions would be:
First, ditch the rifle - if he's going to be a 6 shooter guy he really doesn't need that. His 6 shooting mechanics should be good enough.
On his reload passive, he should just have a stack counter that goes from 0 to 5 (representing a fresh revolver vs only 1 shot left), these stacks can interact with his abilities in meaningful ways to make keeping track of how many bullets you have left more interesting than "welp I can't attack anymore". You can have a special animation that plays when he's fired his last shot, but this delay should only be slightly slower than his normal attack delay; personally I think it'd be cool if instead of reloading, he tossed away his revolver and drew a new one.
And now that you have this cool stack passive, you now have an interesting mechanic to build around - like one ability might let him empty his pistol really fast, so that would reward having a full clip, while he could have another spell that is just a strong shot that is more powerful if he's on the last bullet in the gun "I saved the special one just for you, pa'tner"
Also, does he really need to have a dash? Consider putting something else here, or if he must have one, make it make sense. Like a cool thematic ability might be, "hit the floor" where he drops to the ground and stays there until you click move somewhere else - he will auto-dodge all projectile attacks, and can shoot from down there but can't reload gun while he's on the floor, and has a delayed animation for him getting back up; this is something that is a fairly unique defense while also interacting with his passive (he runs out of bullets on the ground).
tl;dr, a revolver like mechanic can work, but you should build off that mechanic in interesting ways, rather than just making it a penalty for the player.