If your name has changed you can make an application online to change your name on the electoral register at Gov.uk. Alternatively, you can complete a change of name form with your previous and new name and the date of the change. You would then need to provide evidence to support the change of name, such as a copy of a marriage certificate or deed poll certificate (this is unlikely to be required if you make an online application to update your details).
To request a form please contact Electoral Services on 01529 414155 or by email to email@example.com giving details of your address and previous and new surname. You will then be sent a personalised form to complete and return with the appropriate evidence.