Do I need to open a separate bank account for my business or is it okay to have clients pay straight into my personal bank account?
In short, you don't have to have a business bank account. But it makes things easier.
If you are registered at the KvK and want to open a Dutch bank account, the process is almost the same as it is for opening a private account. You should have all of the necessary details concerning your business like your KvK number, VAT number and sometimes even a record of your turnover.