There is a "Save messages" option in the default Messages app. I have not tried it, but maybe the option will allow you to save all messages you want (can individually select them), and then once saved, copy from the Note 5 to a PC.
Update: Looks like this option saves SMS only, so may be limited compared to what you want to do. Once saved, the SMS message will go to the Messaging folder with a filename structured by year, month, day, hour, minute, and second(s). The file is saved in XML format. As Rukbat mentions above, reading XML is not difficult. XML files can be opened with Notepad, a browser, or a variety of other text editor apps.