At times iCloud won't back up properly, here are a few solutions aimed to help those who are experiencing problems on iPhone, iPad and iPod. Also see: How to recover a forgotten iCloud password
How to fix iPhone won't back up to iCloud: Quick fixes on your iOS device
First and foremost you should check that you've got enough iCloud storage left. You'll be allocated 5GB for free, but anything more than that will require you to pay for a greater iCloud storage allocation. To make sure you've not filled up your 5GB (or greater) space, go to Settings > iCloud > Storage > Manage Storage.
If you've got to clear some space, do so by deleting or unchecking the things you don't want backed up and it should resolve your issue.
An internet connection is required for a successful iCloud backup, and if you are experiencing problems with your network provider or Wi-Fi router, it might be the root cause of your issues - therefore, it's worth connecting to a different Wi-Fi network or even resetting your network settings through: Settings > General > Reset > Reset Network Settings.
Being up-to-date is always important and for iCloud can be the deal breaker. Make sure you're on the latest iOS for your device.
A very simple and yet easy fix can simply be to log out and login to your iCloud account by going to Settings > iCloud and taping the Sign Out button at the bottom of your screen. Similarily, restarting your iPhone can often fix many a lot of issues!
Finally, you can also remove a previous iCloud backup from your device - as that can also help the new one work more seamlessly. Go into Settings > iCloud > Storage > Manage Storage then find the backup and hit Delete Backup at the bottom of the page. Also see: How to use iCloud Drive
How to fix iPhone won't back up to iCloud: Use third-party programs
If you are still experiencing problems with iCloud, you can always backup using third-party programs such as AnyTrans - through the program you'll be able to backup all your files and contacts and be able to integrate it with data saved to your computer or iTunes.
To use AnyTrans, first download it and connect your device to your computer. Click on Content to Mac (or PC), select the categories you wish to be backed up and wait for the process to be complete - remember the greater the storage, the longer it will take.