How can I register a Kindle Fire version 7.5.1 in my Amazon account

To register your Kindle Fire:

1. From Home, swipe down from the top of the screen, and then tap More.
2. Tap My Account.
3. Tap Register. If you already have an Amazon account: Enter your Amazon account information. If you do not have an Amazon account: Tap Create Account to set up a new account.

Amazon.co.uk Help: Register Your Kindle Fire

or go to this link and have a read up.
How do register my kindle fire Google Search

To buy content and deliver it to your device, you will need to connect to a wireless network and register your Fire tablet to an Amazon account.
If you are setting up a new Fire tablet for the first time, you'll be taken through setup instructions and will have the chance to register your device. Follow the instructions on your screen.
To register your Fire tablet outside of the initial setup screens:

1. From Home, swipe down from the top of the screen and then tap Settings.
2. Tap My Account.
3. Tap Register.
   • If you already have an Amazon account: Enter your Amazon account information, and then tap Continue.
   • If you do not have an Amazon account: Tap Start Here, and then follow the instructions to set up a new account.

Note: If you've forgotten your Amazon account password, you can reset it by requesting that a personalized "password reset" link be sent to your e-mail address. To learn more, go to Reset Your Account Password .
If you need to deregister your device from an Amazon account, follow the steps above, and then tap Deregister. After you deregister your device from an Amazon account, you won't be able to access items stored in the Cloud for that account.
After deregistering, you can register your Fire tablet to another Amazon account. Related Help Topics

• About Wireless Display Methods for Amazon Devices

To buy content and deliver it to your device, you'll need to connect to a wireless network and register your Fire tablet to an Amazon account. Registering also allows you to transfer purchases between devices and Kindle reading apps. To learn more about connecting to a wireless network, go to Connect Wirelessly.

1. From Home, swipe down from the top of the screen and then tap Settings.
2. Tap My Account.
3. Tap Register.
   • If you already have an Amazon account: Enter your Amazon account information, and then tap Register.
   • If you do not have an Amazon account: Tap Create Account, and then follow the instructions to set up a new account.

Amazon is cautioning clients of more seasoned Kindle tablets that they In the event that the issue continues, if you don't mind restart your Kindle from the Menu in Settings and attempt once more. Arouse fifth Generation. Encourage Touch fourth Generation. Encourage Paperwhite fifth Generation. First gander at Amazon's Fire tablets For more help related kindle devices contact our kindle customer services number +1-844-769-9823.

What you really need to do is add your new Kindle device to your
existing Amazon Kindle account. The good news is if you bought it from Amazon new, Amazon automatically registers it to your existing account. So . . .

Go to the Amazon website using your laptop or desktop computer and log onto your Amazon account.

1)On the upper right of the page you should see *Hello (your name)*
Drop down this menu by hovering your mouse pointer on it.

2) Select *Manage your content and devices*

3) If you're not logged in, enter your Username and password.

4) If you're browser is set to save passwords an log in you won't
need to do step 3

5) Select the *Your Devices* tab

6) All your Kindle devices should be listed there, including the new
Kindle Fire.

7) You can click on your old Kindle and *deregister* it.

8) You can make your new Kindle Fire your *default device* and
you can edit your other Kindle devices such as any Android
Device onto which you have installed the Kindle App.

9) The default device status ensures new books purchased will
automatically download to that device when it's connected to
the internet via Wi-Fi or 3G depending on what your Kindle Fire
is configured with.

10) You can click on the *Settings* tab to manage your credit card and other information.

Registering also allows you to transfer purchases between Kindle devices and Kindle reading apps.

1. From Home, swipe down from the top of the screen, and then tap More.
2. Tap My Account.
3. Tap Register. If you already have an Amazon account: Enter the e-mail address and password associated with your account.

Here on Amazon you should find everything you need to know and a link to register.
http://www.amazon.com/gp/help/customer/display.html?nodeId=201539930

Not sure this will manage all questions in one answer, but you could start here, where Amazon offers support for the Kindle fire and accounts:
Fire Kindle Echo Support

To register your new kindle you need to go to http://www.amazon.com/s?ie=UTF8&page=1&rh=i%3Aaps%2Ck%3Akindle%20registration and follow the directions.

Register Your Kindle Register your Kindle to your Amazon account to buy and deliver Kindle content to your device.

To buy and deliver Kindle content, you'll need to connect wirelessly and register your Kindle to an Amazon account. Registering also allows you to transfer purchases between Kindle devices and Kindle reading apps.

1. From Home, press the Menu button, and then select Settings. Select Registration.
2. Specify the Amazon account you want to use for your Kindle:
   • If you already have an Amazon account: Select this option. Enter the e-mail and password associated with your Amazon account. Select Register. Your name will appear as the Registered User when registration is complete.
   • If you do not have an Amazon account: Select this option and follow the on-screen instructions to set up a new Amazon account.

Having trouble?

• Kindle is Registered to a Different Account

In your Amazon account, ( If you don't have one, create one) simply login and click on the "Manage your Kindle" tab on the right of the screen, and then select the "my devices" option, where you can register or deregister your Kindle.
If your Kindle is second-hand, you'll need the serial number of your Kindle to register it in your name.