You will need to get a Confirmation of Study letter and provide this to your council/local authority.
When will I be liable to pay Council Tax?
You become liable for Council Tax from the day after your official course end date; this is the date on your student record, and appears as your "expected end date" on university documentation and your student ID card. So for example, if you are graduating from King's during the 2020/21 academic year and your expected end date is 4 June 2021, you'll become liable from 5 June 2021, unless you have been granted an extension.
Postgraduate taught (PGT) students with an official end date of 30 September 2021, will be liable for Council Tax on 1 October 2021.
For further information about how your course dates and student status are worked out, please also refer to: