Generally the first step required when opening an offshore bank account or trust is to set up an offshore corporation in a suitable tax haven, or purchase an existing Shelf Company.
International Business Corporations
IBC's are completely private corporations, and are generally allowed to conduct business in any country except the one they were created in. For example, if you created an IBC in Belize, you would be able to conduct business through the IBC in any country except Belize.
This does not stop them banking in the country they were formed in.
Shelf Companies
As an alternative to setting up a new IBC, you can purchase an existing Shelf Company. These are companies that have almost always been registered in a prior tax year. This can be useful if you need an offshore company that you can demonstrate has been in existence for several months or years, or if you need a company that was formed in a prior tax year.
Once you have set up your IBC or purchased a Shelf Company you can then open a bank account for that corporation in the country of your choice, and with the offshore bank of your choice.