The built-in spell checker in Acrobat/Adobe Reader is only meant for use on the text in form fields or comments, not the actual static text of the PDF file.
In Acrobat Pro, you can navigate to Preferences from Edit>Preferences>Spelling and select 'Check spelling while typing'.
Feel free to update this discussion for any further assistance.
As mentioned, the built-in spell checker in Acrobat only applies to form fields and comments.
A script can be used to spell-check the static contents of the document, though.
I've developed a (paid-for) tool that will allow you to do it quite easily. You can find it here: Custom-made Adobe Scripts: Acrobat -- Spell Check Suite