Photos not synching from iPhone 5S to iCloud
Try first by checking on your phone on how much your photos are using up in your iCloud by navigating to: Settings -> iCloud -> Storage xx GB Available -> Manage Storage -> iCloud photo library and there you'll see there amount of photos and videos you have available in your iCloud.
Uploading photos can take a very long time and there lower upload bandwidth you have the longer it'll take and sometimes it might seem like it stopped uploading but really it might just be uploading a long video clip which could sometimes be a few GB long.
What I've also realized at least with iOS 9.2.1 that navigating away from the photos app might cause they upload process to pause so I'd suggest you that if you feel comfortable to remove the auto lock feature on your iPhone that you should turn that off at least for the time that your doing the bulk uploading by navigating to: Settings -> General -> Auto-Lock and select "Never" and then leave your iPhone connected to power with the WiFi connected and within the photo app.
The final verification that you'd want to verify is that you have the adequate amount of storage available on the iCloud account that you're connecting to.