It really depends on what your Wordpress site is about. Are you accepting Bitcoin as donation? Or are you running an online store and wishes to accept Bitcoin as payment?
Almost every new functionality you want to add to WordPress is done with plugins. If you do not wish to use a plugin, then the only way I can think of is to write the script yourself or hire someone to write it. There is no reason to upload a custom script that interfere with the WordPress core if you can install a plugin, do you?
https://wordpress.org/plugins/tags/bitcoinhttps://wordpress.org/plugins/tags/bitcoin-payment