An effective method of characterization is to draw attention to the 'things' the character notices as they move the setting.
The objects that you notice when you enter a room can reflect a lot about your inner state.
For instance, if you come home when you are really tired and exhausted, the comfortableness of a sofa with lots of pillows might be the thing that draws your attention. But, if you are anything like me and you come home angry and filled with rage, you might notice how those pillows fit nicely over some jark-off-mother-duster's face.
And 'things' isn’t confined to physical objects, they can be smells, other people, things from memory likes events from the personal past or historical past. They can even be imagined.
For a character that is going insane they can either notice different things in the settings, or react differently to the same things in the setting. Or they might be making up things or associations with the 'things' as they encounter them.
A lot depends on their insanity -- paranoia v delusions v narcissism v sociopathy.
I think to be engaging and sympathetic, it must start off low and small, kind of quirky and build up, and at times cost the character something.
That said, it depends on the kind of problem the character faces and the story itself, because there are many stories were central characters start off mad and the reader has to kind of play catch up to understand what is going on.
Preparation for a Descent into Madness by Doris Lessing and The Zen of Motorcycle Maintenance by Robert Pirsig and Confederacy of Dunces by John Kennedy O'toole all deal with characters gripped by insanity and portray them very differently.