This should work:
def upload
    render :json => { :attachmentPartial => render_to_string('messages/_attachment', :layout => false, :locals => { :message => @message }) }
end
Notice the render_to_string and the underscore _ in before the name of the partial (because render_to_string doesn't expect a partial, hence the :layout => false too).
UPDATE
If you want to render html inside a json request for example, I suggest you add something like this in application_helper.rb:
# execute a block with a different format (ex: an html partial while in an ajax request)
def with_format(format, &block)
  old_formats = formats
  self.formats = [format]
  block.call
  self.formats = old_formats
  nil
end
Then you can just do this in your method:
def upload
  with_format :html do
    @html_content = render_to_string partial: 'messages/_attachment', :locals => { :message => @message }
  end
  render :json => { :attachmentPartial => @html_content }
end