How do I access pictures stored in the iCloud photo library?

You can use a computer browser (e.g. Safari) and log in to https://www.icloud.com/ and open the Photos app there.

The will also be in Photos on any device where you have iCloud Photo Library turned on. On a Mac it is turned on via Apple menu > System Preferences > iCloud > Photos > iCloud Photo Library. On a Mac it requires Yosemite OS X 10.10.3 or later. See the following for more information: https://support.apple.com/en-us/HT204264

The two methods outlined by FoxFifth are having different limitations.

• The Photos.app on the cloud webpage at www.icloud.com will let you browse the photos, upload new photos, download photos. You just need a browser. But you cannot edit the photos or add metadata, and there is no search function.
• The Photos.app on a Mac or an iPhone/iPad/iPod Touch will allow you to edit and to search the photos. On a mac you can add keywords or titles and description. But you will need an Appel device with the Photos software.