In it's simplest form, for me it's just mayo, chopped or shredded chicken, celery, onion, salt and pepper. I make it a point to try and dry out the chicken as much as possible though because I hate it when too much moisture gets into the mayo causing it to become loose and milky instead of rich and creamy.
Most of the time though, I will also add in some spicy brown mustard, a little sugar to balance out the sourness of the mustard, and perhaps some additional herbs such as chives and dill. I also enjoy throwing in some chopped toasted walnuts, since they go really well with the mayo. Not to mention, walnuts have a naturally occurring form of MSG in them which enhances the flavor of the chicken salad overall.
If you prefer shredded chicken in your chicken salad, a trick I've been seeing recently is to simply add the chicken to your stand mixer, and use the bread attachment on them. It shreds the chicken evenly and quickly, much faster than you could do it by hand. I'm usually more partial to the shredded chicken salad, since I think the strands hold everything together better. I don't like biting into large chunks of chicken salad , which in turn end up falling out of the sandwich I have it in, and getting all over me.