There are a few ways you can go about this:
*Extend the current BizForm webpart and in the OnAfterSave event, perform your newsletter subscription via the API
* Use a global handler. Documentation for v7 and Documentation for v8
I like the global handler specifically because it's not a "one off" scenario, offers a single place for other events to be handled, and gives the developer(s) a single place to go to manage code vs. messing around with creating new webparts and such.