So right now, I have made a package which contains a class that is needed to make a visualforce page on a professional edition account work.
What is the best step to send get that class over to the professional edition account?
You can't put custom Apex code directly into a Professional edition org. Typically this level of customization requires at least an Enterprise Edition Org.
See Salesforce Features and Edition Limits.
The only way to get Apex code to run in a Professional edition org it via publishing to the App Exchange after passing the security review.
The question Building an App that works in Group & Professional Edition? has some other good points about targeting a professional edition org.