How to kill a God?

I’m running a campaign where my players are close to discovering an organization who believes the Gods don’t do enough and aren’t worthy of their power, so they are killing off gods. 

My question for y’all is how would you kill a god? What would need to be done or take place to kill a god? What might the repercussions look like on the world?

These answers partly depend on how gods work in your world, but I assume you've taken the sensible approach of not worrying about it until it became relevant.

The first things that come to mind for whacking gods are "they have some kind of god-killing artifact" and "they have control/connection to some kind of entity that can kill gods". (For the second one, it could well be another god in a funny hat, or it could be something outside the realm of the divine.) A lot of D&D worlds have gods gaining power from their worshippers, and the way to kill them is to wipe out their followers, but that doesn't really work for the narrative you're going for.

As for the repercussions, the most obvious is that the god's worshippers will lose access to their powers. (Note: don't do this to a player without some kind of quickly-available power backstop --  losing access to your character abilities for any length of time is not fun.) 

It's also going to be a source of political upheaval. Religious organizations are going to be movers and shakers in most worlds, and when they lose backing, other powers, both secular and religious, may look to take advantage. They won't immediately lose power, but it'll be a big shakeup, particularly where they're enmeshed in, or are the power structure.

Further consequences are up to you. Do the gods actually make the world run? If you kill Apollo, does he no longer carry the sun across the sky in his chariot? Or does it continue onward? Or are there slower, but still potentially catastrophic consequences, like the fires of the sun starting to slowly fade? 

Also, if they do kill a god, and there are world-altering consequences, there's going to need to be a way to fix it. (You don't have to, and probably shouldn't, have a solution in mind right now, but you need to put some elements that the PCs could use into play.)

My go to is a knife made of cold iron, quenched in the blood of an immortal, and tempered with anti-magic.

Really it depends a lot on the kind of power scale you're writing from. For typical D&D lore, you're looking at something in the tier of Olympians, so realistically mortals have no shot- they're generally unreachable, can readily find and smite mortals who are getting too uppity, and it usually takes special plot devices just to inconvenience them. Obviously this isn't the only model, but if you're doing anything along the classic D&D lines, it begs the question of how they're keeping the gods from becoming aware of them and taking direct retaliation.

The most common handwave in D&D against divine interference is that the pantheon(s) collectively have some code of conduct that limits actions so they don't end up all constantly stomping the sandbox flat, but a common credible enemy would logically be something that'd create enough consensus to get past that- though there is the workaround that the pantheon is in fact so badly dysfunctional/complacent they don't take the threat seriously until they're on the back foot.

Honestly, I tend to find the "gods need prayer badly" trope doesn't really track to D&D- the usual premise is the gods came first, arranged the setting, and then populated it, so why are they now terminally dependent on followers when it took a lot of divine heavy lifting to make them in the first place?

And even minor deities tend to have tens of thousands of followers.