I hiked the northern half of the trail a couple years back. You should expect to spend $1,000/month at a minimum for a good experience. I spent $2,000/month and felt like I was living large. (I'd eat like a pig at every hotel/bar/restaurant I entered when arriving into a town. Most people lose weight on the trail; my weight stayed the same.) Expect your gear to cost around the same as your monthly budget.

I saved low five figures but I don’t expect to need all of that. On very rainy days, or when I need to do laundry, I typically go to a hostel or split a hotel room with fellow hikers. Other than that it’s really just food and miscellany…having said that I am carrying like $1500-$2000 worth of gear at any given time so there is a real startup cost.