If some of the gods kill each other out of view of the players, in little fights, what would happen? What would happen to a cleric?

First and foremost, "Do gods kill each other?" is extremely dependent on the pantheon in question.  Most Earthly pantheons have stories of how one god killed another and it made this or that happen.  The Greek Olympians (and the Romans by other names) came about when Zues killed their father, Kronos, overthrowing the Titans and locking them away in Tartarus, the deepest part of Hades (the underworld).  In Dragonlance, Takesis (?) could only be slain after Paladin (?) renounced his own godhood, to maintain the balance between Good and Evil in the heavens.  In Theros, the mortal Satyr Xenegos became a god, and his ambitions threatened the entire plane, until Elspeth and Ajuani together destroyed him (at the cost of Elspeth herself).  It all depends on how things work in your setting.

In theory, if a god were killed, imprisoned, or robbed of their godhood, then they could no longer enact their will through their followers, so Clerics and Paladins would lose most if not all of their spellcasting ability.  This would, naturally, cripple those classes, leaving any players of those classes feeling like dead weight.  If that's your angle, tread very carefully.  If you do not have that concern, and it is NPCs who would be affected, feel free to go nuts.  Depending on how your world works, perhaps the laws of nature could even change as a result of the god responsible for them being unavailable.

In my campaign world, gods go in for more intricate punishments than just death. For example, Lolth used to be a godess until she betrayed the other gods by interfering directly with the mortal world, so they sent her back in time to the Beginning and cast her down in the Abyss, creating a new timeline where Lolth was always a demon, her godhood all but forgotten, forced to wallow in the corruption of the abyss for the rest of eternity. 

Short answer: If you, the DM, want them to, they do. 
As for what would happen to a cleric, that’s a good question, and would depend on the cleric.  I’d guess they would lose their cleric powers until or unless they swear allegiance to a new god. A lot of pantheons have gods with overlapping domains, so they might be able to transition to an allied or similar god and keep their powers.
I could see some becoming disillusioned and simply retiring from adventuring life. 
Others might start to follow the god who killed their god (I’d let them just swap domains for something like this) 
Still others might swear vengeance and hunt down followers of the murderer god. 
Or they could maybe continue on doing their God’s work, because they still believe in the mission. 
If this is your campaign, you might let the god, just before they die in a last act of devotion to its followers, release a burst of holy energy reaching out to its clerics that would be able to reshape them (in game terms, let them respec their character into a different class). This could dovetail nicely with the avenging sort of former cleric, or the carrying on with the work kind.

The Enuma Elish is pretty much just one long story of the Babylonian gods killing each other.  What impact that has on individual clerics is up to you.  Personally., I've always viewed it as a really good reason to throw the entire concept of D&D religion, as practiced in the Realms, overboard.  

A god fight should be the worst disaster in memory.  Happens maybe once every thousand years or so.

Would a Cleric that works for a killed god take madness from it? 

Seems possible. Though it would depend on the cleric, some might not be as devout, and therefore more resilient. And I’d think there would be a save. 

Depends on your pantheon. I might say that clerics (not paladins they use oaths) could lose power or they might still have power if there god dies. As long as they believe in what the god believed in they might still have power. But what about druids? So a druid is on the Plane of Fire. Not much nature there right? But also if Zeus died there might not be any more lightning or even rain or the storms could rampage over the loss of the god that controls them!

In most D&D worlds, yes, the gods kill each other. Happens all the time in Forgotten Realms, and many other settings, like Theros and Wildemount, have gods dying occasionally. As for the cleric question, up to the DM. I personally would allow a cleric to switch over to another god with a similar domain (like from Jergal to the Raven Queen) in order to keep their power. 

Keep in mind, though gods can be killed, they can be resurrected fairly easily as well. 

Quote from LeviRocks >>

In most D&D worlds, yes, the gods kill each other. Happens all the time in Forgotten Realms, and many other settings, like Theros and Wildemount, have gods dying occasionally. As for the cleric question, up to the DM. I personally would allow a cleric to switch over to another god with a similar domain (like from Jergal to the Raven Queen) in order to keep their power. 

Keep in mind, though gods can be killed, they can be resurrected fairly easily as well. 

 this, gods kill gods. mortals kill gods. gods become mortal. mortals become gods.  its a pretty crazy mess.  i definitely wouldn't do anything like that in an adventure though...although trying to kill a god and failing might work.

If you intend to have this sort of thing happen, I would as a DM be part of the story, maybe the player has a premonition and he has suddenly a quest to save his dying god. Maybe it's a really hard quest and the sucess factor is slim (if your endgoal is to have a good chance of what is it like for the player to live through the change of having to "switch" their fate onto a new diety. 

This way, you give a chance for the player to have a hand in the result of this. Maybe they heroicly save the day or maybe they fail and then the journey to explore new option starts. In one of my campaign, I had a player lose their connection to their diety temporarily and it became a personnal journey to finding their faith again. it made for some really interresting roleplay sequences and afterwards, the player had a stronger connection to their characters and what it means to be a cleric. regardless of the outcome, the player will be a part of the story (which in my opinion) is always better then being a spectator to their own story. Plus either scenario makes for a amount of sessions depending on the level of difficulty and complexity of it all. 

Maybe they are witness to the killing blow and then need to do something to bring the god back or release it from whatever fate they were dealt. It,s tempting to have this sort of thing just happen outside of player control, but they will likely feel forced into a situation and might not have the same involvment with the story. Also it might make the player indifferent to the new diety and basically veer them away from understanding what makes a cleric a cleric.

Gods/entities that affect player's classes are always great means of moving your story along and also instantly involves charaters and players along for the ride. 

As an example, let's say your character tries to save their god, but fails for it is an almost impossible task, yet a tiny spark of the god's power/essence lives through this character because they tried and showed their faith to this entity. Now the player has a closer connection to their god, but cannot weild the power of the cleric (which is fine for a few sessions, the rest of the group will pull through) and now he/she hears the god's whispers and it becomes a quest to bring the spark to the halls of "insert pantheon here" and as such, the players explore this avenue and then they restore the spark, restoring the Cleric's powers as well, maybe even a slight boon (homebrewing stuff for special quests is always a plus for the player)

Have fun with it, you could also discuss this with your player and ask them about how their relation with their gods affect the way they see their character. 

also, you might want to review the Forgotten Realms Style Guide - its your adventure so do what you want, but I think its pretty good advice (you didn't say your adventure is in Forgotten Realms, but I think this is good advice if its in any official setting). If you have a huge gigantic event in your campaign and then every official thing that comes out after that contradicts what happened in your adventure, it kinda dilutes what you did, ya know?

Replace Your Divots
Stories should leave the Forgotten Realms as they found them. (Pack it in. Pack it out.) Stories set in the Forgotten Realms should not definitively destroy or irrevocably change locations, people, gods, planes, races as a whole, or other established elements of the setting or brand without special dispensation. That doesn’t mean that stories can’t have high stakes with important things in jeopardy. Players of a D&D game have the option of failure or of causing the death or destruction of important elements of the IP, but there must always be the option of things being restored to order.

you can find the doc in two places:

https://adventurersleague.files.wordpress.com/2018/10/1765690-forgotten_realms_style_guide.pdf

https://www.dmsguild.com/product/267467/DMs-Guild-Creator-Resource--Style-Guide-Resources

Yes, gods do die. In Forgotten Realms, Cyric killed Mystra which made the Spellplague happen. What happens to a cleric, I'm not sure. Because Kua-toas get powers simply in believing, they might still have the power. It's up to you, the DM, though.