Sitting for a long period of time in a worn office chair that's not well-suited for comfort or posture may cause health problems. Sitting for long periods can put excess stress on the lumbar region of the spine. Sitting in the wrong chair can cause back problems, headaches or wrist pain from having to incorrectly reach or lean to your work area. Ask your boss to provide you with a new work chair to improve your workflow or offer to purchase one on your own if she agrees to reimburse you.
Research the relationship between office chairs and back pain or whichever relevant ailment you're suffering. Not liking the looks of your chair won't likely be a good enough reason for your boss to replace it.
Arrange a meeting with your manager or the person responsible for ordering office furniture.
Meet with your manager and discuss your need for a new office chair. Explain why and offer your research to support your claim without sounding forceful or threatening. For example, explain that you've been having some pain in your wrist for the last few weeks and believe it's because you can't properly reach your work area.
Wait for a response from your manager. If she sounds hesitant, offer to purchase one yourself if she agrees to reimburse you. Ask for a price limit if your manager agrees.
If your employer is short on cash, consider shopping for a used chair instead. Let your manager know you're open to a slightly used chair for a lesser price. Research classifieds and give your manager a price range.