Originally I was going to make this topic asking for some help with killing shopkeepers until last night when I got the Public Enemy Achievement. So instead, I figure I would show what I've learned after countless deaths trying to kill 12 of those lethal shopkeepers with the hair trigger and dead aim with a shotgun. It wasn't easy, that's for sure.
Number One Tool, the Shotgun
I started this task with the idea that the shotgun was going to be the best bet to kill shopkeepers, and for the most part I was right. Due to the random nature of Spelunky for level generation, getting a shotgun can be tricky. The easiest way is when a shop has the shotgun on sale. You can easily pick it up (as if you were going to steal it) and blast the shopkeeper point blank. Be careful, because even that's no sure thing. I've had a few unlucky runs where the shopkeeper has seemingly survived long enough to one-shot me with his own gun. Not to mention he jumps around like a lunatic and can melee you to death at will. I suggest you keep shooting until you no longer see stars hanging over the shopkeeper's head. It's perfectly okay to go overboard. It's safer that way.
The second easiest method is finding a shopkeeper trapped in a tiny room with treasure chests. It's recommended you try this only if you can get above him. That way you can bomb a pit where he can't escape and kill him with an explosion while he futilely jumps around. Of course, you'll need two or three bombs to get in there and kill him and some ropes to get out of that pit once you grab his shotgun. I haven't managed to pull this off from any other angle as the shopkeeper eventually escapes and kills me. Maybe you're way better with bombs than I am, but I think death from above is the only option. If you can't get in there safely, it's best to leave him alone.
The last option (and most desperate I find) is attacking a shop and hoping you can lure the guy into a trap or catch him in an explosion before he kills you. Usually this method ends in failure every single time for me. The few times I've pulled it off is when the shopkeeper jumped into a spike trap or was killed by an enemy. But it's extremely dangerous and not worth the risk.
Also, killing shopkeepers will give you more shotguns than you'll know what to do with. I'm not sure if other weapons work better, but you won't have to worry about running short of guns this way.
Other Helpful Things, Tons of Bombs + Sticky + Spring Boots
The shotgun is a nice start, but it's better if you can grab yourself tons of bombs. The nice thing about killing shopkeepers is that you can loot their store freely when they're dead. Sometimes you can get lucky with a bomb shop and get out with 20+ death dealing explosions for free. You'll also want sticky bombs (easily found on dead giant spiders) considering shopkeepers are so fast and jumpy that they'll normally avoid live bombs unless they're stuck and have nowhere else to go. Sticky Bombs make sure that if you get a good throw in and hit them, they'll die no matter where they run. And you'll want to take your shots carefully because even 30 bombs can go quickly if you're desperately trying to kill a rampaging shopkeeper somewhere below you.
Spring boots are also key because you'll have more mobility and opportunities to escape if things go bad. You could try the jet pack if you find it, but I found the jets were too slow to escape the high jump shotgun blast of the shopkeepers. In the thick of a rampage, you'll want to move as fast as you can and running and jumping worked best for me.
Best Angle of Attack, Edges of the Screen
For lack of a better term, a shopkeeper's cone of vision doesn't reach across the whole level. You can keep them just out of sight on the edge of your screen and blast away with impunity. You can't see your shotgun bullets hit, but they keep going until they hit a wall. You have to use that range to your advantage because if you get in close and the shopkeeper starts rampaging, you're most likely going to die because your controls are nowhere near sensitive or precise to match your crazy target. I recommend you keep firing a volley of blasts before moving in to see if your target is dead or not. Shopkeepers can survive numerous shotgun blasts if you're not close enough. I've died many times thinking I've killed them when I only dizzied them.
Another good opportunity to attack shopkeepers is a lot like the treasure room strategy from above. If they do jump into a narrow shaft where they can't escape, feel free to rain bombs on them until they die. If you have Sticky Bombs, this tactic is a little harder to pull off because if you hit DOWN + X, the bomb will just stick where you're standing and most likely give the shopkeeper a way to escape. You'll have to jump a bit and throw downward and get lucky.
I recommend against chasing or jumping down after a shopkeeper even if you're armed with a shotgun. Unless you're quick in a tight situation or have good aim for jumping on people's heads, the shopkeeper is fast enough to recover, grab his gun, and end you in one blast. Watch yourself around ledges.
Watch Those Exits and Unconfirmed Information
A key thing to remember, once you've pissed off a shopkeeper, one of them will always patrol the exit waiting for you. I mention that because I've forgotten about it a few times and gotten too greedy looking for kills when I could take it easy and rack them up through level progression. I stress that you choose your shopkeeper targets wisely. They are incredibly dangerous and the ones around the exits are usually easier targets because you often have line of sight where you can just fire away until they die. Or in the mine stages at least, sometimes they're sitting ducks in cramped areas where you can just drop a bomb and kill them easily.
The unconfirmed part is how much of a role you have to play in the death of a shopkeeper. From what I experienced, I think they just have to die period. A couple of shopkeepers jumped to their deaths in the Ice Caves stage when they went on a rampage and I still got credit for their deaths. So if a shopkeeper gets stabbed from a spike trap, eaten by a killer plant, or plunges to their doom in an abyss, I believe that counts to the achievement. So don't panic if you didn't get in the final shot or witness them die at someone (or something) else's hands. Just keep rolling until you see the message pop (or get killed trying).
Good Luck, Have Spelunky (Forget Damsels)
Of course, on top of everything is the normal run of Spelunky luck and enemy placement. If you ever get a stage that is "too dark to see" I say forget it and start over. You can try to navigate the darkness, but shopkeepers don't have the same handicap. Not to mention you only have 15 stages to find and kill them before the climax at the end, so every kill counts. As for Damsels and treasure, most shopkeeper killer runs are essentially suicide. An extra point of health won't save you from a shotgun blast if he gets in close. I mean, unless the damsel is painlessly close to the exit, sure. But don't try to escort them through a level and juggle a shotgun at the same time. It's not worth the trouble.
The nice thing is that the further you get in the game, the easier it is to kill shopkeepers. The levels get more deadly and rampaging shopkeepers have less room to jump around and move before falling down an pit, impaling themselves on spikes, or leaping into lava. All you can do is keep trying until you get results or call it quits. So good luck, have Spelunky.