Backing up your iPhone to iCloud is most likely a lot easier than you think. You basically only have to have an Apple ID, set up iCloud, and then flip a switch in settings to enable automatic backups. To do this, sign in to iCloud through Settings > iCloud. Then, go to Settings > iCloud > Storage & Backup > iCloud Backup = On. This will turn on the magical switch that will back up your iPhone to iCloud.
If you have any questions about this method, please post below!