Q: Can't backup nor delete iCloud

Hello there, papigiulio.

It sounds like you're trying to log yourself out of Find My iPhone with your old apple ID so that you can restore your iPhone backup to your new iPhone 6s. As long as you still remember your password, you can use the email address you had for your Apple ID to login to iCloud:

iCloud: Remove your device from Find My iPhone

Remove an iOS device or Mac by turning off Find My iPhone

On an iOS device: Go to Settings > iCloud, then tap to turn off Find My [device]. 

On a Mac: Choose Apple menu > System Preferences, click iCloud, then deselect Find My Mac. 

Note:    You can also remove your device by turning off iCloud completely on that device. Turning off iCloud turns off all iCloud features on the device. 

Remove an iOS device or Mac on which you can’t turn off Find My iPhone

If you can’t turn off Find My iPhone on the device, turn off the device so it goes offline, then remove it from Find My iPhone on iCloud.com. 

Note:    You can also remove your iOS device by first erasing it—just follow the instructions below for removing an iOS device you don’t have. You can later restore the device from an iCloud or iTunes backup. 

1. Turn off the device you want to remove.

2. Go to Find My iPhone on iCloud.com. 

    

   If you don’t see Find My iPhone on iCloud.com, your account just has access to iCloud web-only features. To gain access to other iCloud features, set up iCloud on your iOS device or Mac. 

3. Click All Devices, select the offline device, then click Remove from Account. If you don’t see Remove from Account, click All Devices again, then click the Delete button  next to the device. 

    

   If the device comes online again, it reappears in Find My iPhone. If your device reappears, turn off Find My iPhone on the device (follow the instructions above for removing a device by turning off Find My iPhone), or if it’s an iOS device and you no longer have it, follow the instructions below for removing an iOS device you no longer have. 

Remove an iOS device you no longer have

If you no longer have the iOS device because you gave it away or sold it, you need to remotely erase it before you can remove it. 

1. Go to Find My iPhone on iCloud.com. 

    

   If you don’t see Find My iPhone on iCloud.com, your account just has access to iCloud web-only features. To gain access to other iCloud features, set up iCloud on your iOS device or Mac. 

2. Click All Devices, then select the device.

3. Click Erase [device], then enter your Apple ID password or your family member’s Apple ID password. Because the device isn’t lost, don’t enter a phone number or message. 

    

   If the device is offline, the remote erase begins the next time it’s online. You receive an email when the device is erased.

    

   If you erase your iPhone 6 or iPhone 6 Plus and you have credit and debit cards in Wallet for Apple Pay, Find My iPhone attempts to remove your cards immediately, even if your iPhone is offline. For more information, see the Apple Support article Use Wallet on your iPhone or iPod touch. 

4. When the device is erased, click Remove from Account.

    

   All your content is erased, and someone else can now activate the device. 

You can also remove your iOS devices, Mac, or Apple Watch from your iCloud account. For more information, see Manage your iCloud devices.

One thing you can do once you get your restore in order, is change your Apple ID to reflect your new information. This article will help with that process:

Change your Apple ID

Alternatively, if you've forgotten your Apple ID password or answers to your security questions, you may find this helpful:

If you forgot your Apple ID password

Thanks for reaching out to Apple Support Communities.

Kind Regards.