You would be the person benefiting from the care and the funding, so the court judgment looks like it was made against you directly. The local authority simply fund the care so the claimant is correct in pursuing you.
For as long as the CCJ is in place, and remains unpaid then the claimant can enforce with bailiffs. It also stays on your credit record for up to 6 years. As such it needs to be set aside.
To set aside the CCJ you need the N244 form and a witness statement with a draft order. I attach copies of those which you can tailor to your specific case. With the witness statement, if you want to refer to any evidence such as email correspondence, letters and so on, you can do that and attach them to the statement.
You will need 3 copies of the bundle (N244, statement, draft order) and a covering letter to the court with a fee of £255 payable to HMCTS. If you are on low income or have low savings (or in receipt of benefits) you can ask the court for a fee exemption - so instead of sending payment you enclose a completed form EX160 (copy attached).
You can specify a telephone hearing and you ask the judge for an order that the CCJ is set aside. Assuming they agree to that, it comes off the credit record and you will be allowed to either defend the claim if the debt is disputed or alternatively pay what you owe (the local authority should pay) and there would be no mark on the credit record.
If you have not received any court details (you need the claim number for the purpose of the N244) then you can check the CCJ details at http://www.trustonline.org.uk )
If you want to do the application and send me a copy to check, I would be happy to do that. Bear in mind the court expects these applications to be made promptly. If you want an advocate to do the hearing for you, you can use http://www.lpc-law.co.uk who would charge around £250-£350 and the fee is recoverable from the claimant if your application succeeds.
I would go with the advocate as if your application succeeds (and I think it will) then their fee is recoverable from the other side - in addition to the £255 application fee.
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