I have made several requests for special equipment over the years, some granted and some not. Here's what I did to (as far as I can tell) maximize the chances of success:
Ask. I can't tell if you've asked outright or just hinted at it, but don't rely on hints.
Explain why. Your chair is causing you pain; this isn't just a preference. Be prepared to explain how you know it's the chair and not some other aspect of your setup, but you probably won't need this detail.
Do the leg-work. Don't just say you want a new chair; come with a few options of chairs that have the feature you need that your current chair lacks. (For me, being able to lock the back of the chair in place, so if I lean back I meet resistance rather than my chair just falling back, is important. Our chairs didn't have that.)
Be budget-minded, but use mainstream vendors. If your company is going to buy it, then if they have any financial/accounting processes at all (they're not just a few guys in a garage) they're probably going to need to obtain it new from a known vendor. That used chair you found on Craigslist for $10 might not actually line up with their processes (but see below).
If your boss can't help you, try HR. HR probably has processes (and perhaps budgets) for this separate from what your own department has. And if the problem is such that there's mandatory compliance in your locale, they'll be more familiar with those rules than your manager is.
Finally, if they say no, just buy it yourself. Label it as your property and take it with you when you leave. (Some employers have strong opinions about design/appearance, so it's a good idea to ask first.) The $100 I spent on a chair at my last workplace was more than worth it, weighed against 8+ hours per day for years. If you actually can solve your problem for $10 on Craigslist, just do it and save your management requests for things that are more expensive.