H.P Lovecraft is kind of the master for first person madness narrative, so I'd recommend you check him out too.
If you were going for a third person perspective, consider playing with expectations. With words, either in dialogue or in character thoughts, what's happening. Set things up to seem a certain way from that character's individual perspective. Maybe show other characters around them reacting differently, signaling that someone is interpreting something differently.
Madness can also be defined in character motivation. Would a religious zealot purging the unclean be considered mad? I'd say yes. Madness is not just in irregular thought, but also irregular action.
But, if we go by definition, you'll see multiple possibilities from mental defect, foolishness and/or crazy, insane.
Aside from that, I'm likely going to have to research it. Whether it's doing medical research or studying other novel in handling it. This is not a character type I've really written about. So yeah, I would have my research cut out for me.
False premises. From a tiny acorn doth mighty oaks grow. So does the madness when one proceeds from a single, fundamental error. If eternal peace, prosperity, happiness and freedom from want could result, if such and end was real and tangible and true... to what ends would one go to reach for that Earthly paradise? History is rife with lesser examples, lacking in foundational proofs and riddled with corruption within. On the other hand, what if you could prevent a great vileness, an evil incarnate from growing did you but do one small thing to stop it... When would you stop, if you were convinced of that the horrors were true?
Persecution. All manner of harms and misfortunes visit us mere mortals on this Earth. Some greater, some lesser. What if you had proof that your various injuries were not merely happenstance, the ill luck that graces us all- but the result of a malevolent intent? What if proof, valid, believable, testable proofs were offered you in secret, such that you knew that you were being targeted for some strange reason? And what if no one believed you? How far would you go to reveal the truth?
Other Worlds. What if the reality of our own meager perception were not the limit? Beyond the conscious mind might lay paradise- or entire worlds full of monsters. What would you do if somehow, some way you became able to see and sense these monsters? If you could but know the terrible creatures that lurk beyond what science and observable, testable facts reveal, do you honestly think that you could remain psychically intact, once the things learn of your unnatural perception?
In other words, having periods of time in the story that are exclusively from the 'insane' character's perspective before changing the perspective. This not only would allows the readers to understand the insanity of the character, but it would also show exactly how easy it is to fall into delusion.
In other words, the very real psychological horror that insanity is just as insidiously logical as sanity for those affected.
Other than that, I echo some of those above me, specifically obsessional and delusional. (Maybe an unclear example, but what comes to mind is a video game character from the Fire Emblem series—whose sole motivation is to kill those responsible for his family’s deaths, but still kills the people who aren’t related to that incident just because they’re on the same side as the person that is.)