Photo stream doesn't add the albums on your phone because the albums do not contain the actual photos. Albums only contain links to the actual photos in the camera roll (much like a playlist in iTunes doesn't contain duplicate songs but merely links to the songs in your library).
Also, you can't use iCloud to back up your photos. Photo stream photos only remain in iCloud for 30 days, and if you delete them from your phone, they are also deleted from iCloud. Camera roll photos are included in your iCloud backup, but if you delete the photos from your phone, then continue to back it up, the backup containing the deleted photos will be overwritten by a backup that doesn't, and they will be lost.
To keep your camera roll photos, import them to your computer using your usb cable, as explained here: http://support.apple.com/kb/HT4083. If you need to back up photo stream photos that aren't in your camera roll, save them to the camera roll by tapping Edit (Select in iOS 7), tap the photos, tap Share (the share icon in iOS 7), then tap Save to Camera Roll. Then they can be imported to your computer as well.