How to access ~/Library from network booted OSX
"~/Library" is the short path for "/Users/UserName/Library" where "UserName" is the name of the user.

"/Users/UserName/Library" is the short path for "/Volumes/VolumeName/Users/UserName/Library" where "VolumeName" is the name of the volume.

"/Volumes/VolumeName/Users/UserName/Library" is the full path of a user library.

If you want to access a user library from a different volume than the start / boot volume, then you must use the full path.

Select "Go To Folder" in the "Go" menu of the Finder, enter the full path and confirm.

Please note that you might get a permission error if you (aka the current user) do not have access to this folder.