Have a look at this. Scan computing. This was just a quick component put together and of course you can tweak it. I have used one of these cases and are very quiet. There is also no need of a separate CPU cooler as for your purposes the stock cooler is fine.
To keep the price within your budget I have selected Windows 7 OEM 64BIT hence 8GB RAM.
This is £11 over your budget and you may need to add thermal paste if you cannot get the small amount you need elsewhere.