For iCloud to backup your iOS device, all it needs is an Internet connection. So if your iPhone is connected to the Internet via your Router or Airport Express, there's no reason why it shouldn't backup.
The backup only occurs when the device is connected to the Internet via Wi-Fi, when it is connected to a power source and when the screen is locked.
I understand that but in this instance that is not the case. There is extensive proof that the Arris TG862 router does not allow IOS devices especially the I phone 5 the ability to backup to icloud. So, I put the Arris into bridge mode, purchased a aiprot express to go around the arris TG862 hoping it would solve the problem which it did not. I am confused and do not know where to go from here.
Hmmm.. I'm not sure then.
I just had a quick Google around and found the following thread which suggests a couple of potential fixes for using Apple devices with that router.
If you haven't already, I guess it wouldn't hurt to try those.