If I die, does council tax die with me?

No

When a person dies, their debts are paid out of their estate.^[1]

If the debtor made a Will, the executor is responsible to applying for a grant of probate,^[2] or if there is no Will, the next of kin applies for Letters of Administration.^[3]

If a bailiff seeks recovery of a debt by taking control of the debtor's goods, the enforcement power ceases to be exercisable against the debtor because the goods of the deceased become vested in their estate.^[4][5]

A creditor applies to the debtors executor or administrator for the payment of debts from the residue of the estate.

If the estate does not have funds, the debt dies.

^{[1] Section 32 of the Administration of Estates Act 1925
[2] Section 7 of the Administration of Estates Act 1925
[3] Section 9 of the Administration of Estates Act 1925
[4] Section 32 of the Administration of Estates Act 1925
[5] Paragraph 6(3)(c) of Schedule 12 of the Tribunals, Courts and Enforcement Act 2007}