It is not API stuff in that way.
API is a means of commication with 3rd party systems who will also have API. You create the middleware to link the two together.
This can be any programming language you like using that API (SOAP) so in what I know you use - PHP
DW supports PHP syntax so you can have your host and code what you need to do to make it work with BC.
See my other post regarding PHP integration. I've posted a simple implementation of the PHP Soap client. You can edit that in Dreamweaver if you wish, but DW is just an editor for PHP files. You can use any editor you wish, but you still have to host your code on a machine with a web server. If you're on Mac you will have Apache on it alread, which you can enable through System Preferences > Sharing > Web Sharing. You'll have to install it on Windows. For development purposes (not production) you can also use some of the small Python servers which can run in your terminal on a port you specify and can be easily restarted.
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