You need to obtain a letter from the other party to confirm you have paid them in full. You can then apply to the court for a Certificate of Satisfaction, which they will issue for a small fee (currently �15).
The Credit Reference Agencies' files will automatically be updated within a month or two of the certificate being issued, or you can contact them directly if you want to speed it up a little.
Their files will be updated to show the CCJ as satisfied. It will remain on your credit record for six years from the date of satisfaction, but this is better than it continuing to show as unsatisfied.
As an aside, had you paid in full within 28 days of the judgment, you would have been entitled to have the CCJ removed from your credit file. Rather than a Certificate of Satisfaction, you would have been issued with a Certificate of Cancellation. This would have removed any trace of the judgment from your credit file.