He's willing to kill Flint in order to take something that he has no right to take.
Hmh? The local points a gun at Kirk; Kirk tells him he's now fully within his rights to gun down the asshole and then throw the corpse in jail. Sounds like an everyday situation in any country where possessing a gun isn't illegal to begin with.
Kirk then adds that with Flint justly killed, Flint's possessions will also be free for taking, so it's a lose-lose situation for the stupid criminal. Flint backs off.
No "unnecessary killing". Merely a just threat of force for self-defense. By a de facto officer of the law, no less. Kirk really showed considerable restraint there.
Kirk and company were trespassing. Flint told them so. M-4 fired a warning shot yet Flint specifically said not to kill them. When Flint told them to leave his planet Kirk refused.
Captain's log, stardate 5843.7. The Enterprise is in the grip of a raging epidemic. Three crewmen have died and twenty three others have been struck down by Rigelian fever. In order to combat the illness, Doctor McCoy needs large quantities of ryetalyn, which is the only known antidote for the fever. Our sensors have picked up sufficient quantities of pure ryetalyn on a small planet in the Omega system. We are beaming down to secure this urgently needed material.
(Compared to the lurid red, purple and vivid blue of the planet from space, the ground is pretty normal)
MCCOY: Jim, there's a large deposit bearing two seven three, four kilometres away. I've got four hours to process that stuff, otherwise the epidemic will be irreversible. Everybody on board the Enterprise will
SPOCK: Strange. Readings indicate a life form in the vicinity, apparently human. Yet ship's sensors indicated this planet was uninhabited.
KIRK: Let's get that ryetalyn.
(A strange device comes travelling through the air towards them. A bit like Nomad, but much smaller and with more round bits. It fires an energy beam at the landing party's feet. They try to fire their phasers.)
(The device fires again, and closes in.)
FLINT: Do not kill.
(The device backs away and a silver-haired man in sort of doublet and hose approaches)
KIRK: I'm Captain James Kirk
FLINT: I know who you are. I have monitored your ship since it entered this system.
KIRK: Then if you know who we are, you know why we're here, Mister?
FLINT: Flint. You will leave my planet.
SPOCK: Did you say your planet, sir?
FLINT: My retreat from the unpleasantness of life on Earth, and the company of people.
KIRK: Mister Flint, I have a sick crew up there. We can't possibly reach another planet in time. You can't refuse us the ryetalyn.
FLINT: You're trespassing, Captain.
KIRK: We're in need! We'll pay for it, work for it, trade for it.
FLINT: You have nothing I want.
KIRK: But you have the ryetalyn that we need! If necessary, we'll take it.
FLINT: If you do not leave voluntarily, I have the power to force you to leave or kill you where you stand.
KIRK: Kirk to Enterprise. Mister Scott, lock phasers onto our co-ordinates.
SCOTT [OC]: Aye, Captain, all phasers locked on.
KIRK: Mister Flint, if anything happens to us, four deaths and then my crew comes down and takes that ryetalyn.
"You can't refuse us the ryetalyn." Oh really? So Starfleet is entitled to whatever they feel they have a need for? The Federation talks a good game about respecting rights but when push comes to shove they rely upon their so called defensive weapons to force their will upon others.
Flint did not cause the plague nor did he prevent the ship from leaving. He asked them to leave and they refused. Is he not entitled to defend his home? Is be being an asshole? You could certainly make that case but there's no law against that, especially since he's not a Federation colony. And why didn't they check the computer for information on the planet before beaming down?
KIRK [OC]: Just a little while longer, Scotty. Report on the computer search.
UHURA: There is no report on Mister Flint. He doesn't seem to have any past.
UHURA [OC]: The planet was purchased thirty years ago by a Mister Brack, a wealthy financier and recluse.
They would have realized that the planet was privately owned before trespassing. Note that it's a computer search, not a message from Starfleet. They had the information available, they just couldn't be bothered to look it up when they first arrived.