Cant give you specific suggestions for your budget, but what I can do is suggest you look at a computer chair as an investment and get a really good and expensive one so you dont have to buy a new one for many many years. If you spend a lot of time sitting in it, and you also say you want good back support, you should look at higher quality ergonimic chairs instead of "office supplies store" -chairs.
Last year, I spent quite a lot of money on my new chair, and it has all the support and settings in the world, and I havent looked back since. Granted, I had to save up for it, and I spent more than anyone I know would on a chair, but its something I know is good for my back, its comfortable and something I can have for the next +25 years.
So my advice is to look at much higher quality than you normally would. It pays off in the end