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?
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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.
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.