You can't actually remove any data from your credit report (unless there are inaccurate entries and there is a process for that).
However, after 6 years, the CCJs and all other information will 'fall off' the report. Unitl then, they will remain on your file. If you have paid the CCJ, then it should be marked at satisfied. If you pay it within a month of getting it, then you can apply to have it taken off the register.
You can apply at anytime for your report from www.experian.co.uk and www.equifax.co.uk (there is another one call callcredit but to be fair not a lot of companies use it).