If your coffee bag is made of paper then you can recycle it. If the paper is coated with a thin layer of plastic film, the bag will need to go in your rubbish bin. This is because the composite materials can't be separated for recycling. Coffee bags made out of plastic or metallised film also can't be recycled yet.
You could try buying coffee in bulk online to reduce your packaging waste.
Terracycle offer a paid coffee bags recycling programme. Check it out here.