how to back up imessages

Is this what you're referring to?

Here’s what iCloud backs up:

• Purchase history for music, movies, TV shows, apps, and booksYour iCloud backup includes information about the content you have purchased, but not the purchased content itself. When you restore from an iCloud backup, your purchased content is automatically downloaded from the iTunes Store, App Store, or iBooks Store. Some types of content aren’t downloaded automatically in all countries, and previous purchases may be unavailable if they have been refunded or are no longer available in the store. For more information, see the Apple Support article iTunes in the Cloud availability by country. Some types of content aren’t available in all countries. For more information, see the Apple Support article What can I buy from the iTunes Store in my country?.
• Photos and videos on your iOS devices, unless you turn on iCloud Photo Library beta on your iOS 8.1 device. If you turn on iCloud Photo Library beta, your photos and videos are already stored in iCloud, so they aren’t included in your iCloud Backup.
• Device settings
• App data
• Home screen and app organization
• iMessage, text (SMS), and MMS messages
• Ringtones
• Visual Voicemail

That when you backup your iPhone thru iTunes. The won't show up at the iCloud.com site. There are apps that can read and extract the messages, emails, etc. if the backup is done on the computer. Otherwise you'll have to use the Restore Backup option in iTunes and restore the entire backup.