The most important thing isn't the price range, but what you actually need - and I'd even partially disagree with some advice out there. Sure, you won't get premium ergonomics for €150, but it depends on how long and how intensely you're sitting. If you're really in your home office 8+ hours daily and your back already has issues, then yeah, you should invest more. But it doesn't have to be the €400-600 range. A good middle ground would be around €250-350, where you'll find solid chairs with proper lumbar support, adjustable armrests, and a sturdy gas cylinder that'll last a few years.
My trick: check the return policies. Especially with Amazon and big furniture stores, you can order, test it for two weeks, and return it if it doesn't work out. That's more important than blindly going for the most expensive model. Don't focus as much on "gaming design" - that's usually just styling - and pay more attention to: Can I adjust the seat height? Is there an actual backrest (not just fabric over plastic)? Are the armrests adjustable? A simple, unspectacular office chair for €250-300 will do you more good than an expensive gaming chair if it doesn't fit your body.
But here's the most important thing: go to a store beforehand and try out different chairs for an hour, sit in them, feel how they feel. Then you'll know what your back specifically responds to - whether you need more lateral support, higher or lower armrests, softer or firmer padding. Ordering online is convenient, but with chairs it's really worth doing the in-person check.