how do I determine the size of my photo library

Depends on what you mean and... you can't store an iPhoto Library in the Cloud.

If you use the size determination in the iPhoto Window, that reports the approximate size if you export everything at the 'Current Setting' (which you would not want to do. See below*)

If you check the size of the package in the Finder (Pictures / iPhoto Library) that reports the disk space used for the combined size of the Masters, Previews, Thumbnails, Caches, Database and metadata files.

Though, as I say, it's not really an issue as you can't put an iPhoto Library in the Cloud.

*This User Tip

https://discussions.apple.com/docs/DOC-4921

has details of the options in the Export dialogue.

It is not possible to store an iPhoto Library (or Aperture Library) in iCloud. An iPhoto Library needs to be on a locally mounted volume. If you move the library to iCloud Drive, it will be corrupted, when iCloud updates the local copies on other devices.

See: Use locally mounted Mac OS X Extended volumes for your Aperture library

But to answer your question: You can see the size of your iPhoto Library, if you look at it in the Finder. By default you will find the iPhoto Library in your Pictures folder.

there is no way to backup anything from your Mac to iCloud at this time - it only provided for backup for IOS devices not for the Mac

You can backup photos to Flickr or other photo services

LN

my picture file

What do you mean by your "picture file"?