How can I automatically save site's copy each week with an option to browse saved copies? I need kind of WebArchive, but on my local computer.
You could try using a program like HTTrack to perform a site mirror. This gives you a local browsable copy that you can archive.
You can backup your website easily within the panel of your hosting account. After that, you can download the backup and, thanks to a server engine like Apache, you can browse it locally.