The actual Court Fee for taking your case to Court is £255.
Now, it is up to you if you want legal representation moving forward. Depending on what legal assistance you seek, how many hearings you have and the complexity of your case I'd say £10,000 in legal costs is realistic. For the solicitor to estimate £6-10K I'm guessing your case is fairly uncomplicated as many people who are represented pay at least twice that, maybe even three times as much if their case goes all the way to a final hearing.
If you decide to proceed as a Litigant in Person, you would not have any legal costs other than the £255 (paid by the applicant). There are also other options where you do the majority of the work yourself and tap into legal help as and when you need it i.e. Direct Public Access Barristers or Unbundled service from solicitors.