If you feel confident to represent yourself then it should not cost you more than the court fee. If your ex instructs solicitors/barristers then she will have to pay their fees.
The courts will look at degree of attachment and commitment shown by you to your child over the years and if they are satisfied you are a good enough and committed father, they will grant you the PR even if mother does not agree. They will also want to know your reasons for applying for PR. I would advise you consult with a qualified solicitor before proceeding with this, maybe for a one off consultation, but from the facts you have provided, it seems - in my opinion - highly likely you will obtain the parental responsibility from the courts.
Just remembered another point. If your ex tries to say you are not the biological father to your child and that is why you should not get parental responsibility, then the judge may order that you both pay the costs of a DNA test to confirm this. Although I am sure this will not be an issue as you were registered on the birth certificate? It is probably best you discuss this with your ex first and try to get her to go into a PR Agreement with you first before going to Court.
It seems a shame to involve the courts after 5 years. Good contact for children relies on parents working together . Going to court tends to leave parents feeling resentful and resistant making working together difficult. Even if the mother is dismissive a few sessions of mediation may help diffuse the situation so constructive progress can be made.
In any event before you apply to court you (or a solicitor on your behalf) needs to write formally and ask the mother to sign a Parental Responsibility Agreement. The letter can then be used in court if necessary as evidence that you have been reasonable and tried to resolve the matter between yourselves before applying to court.
If your ex refuses you would need to apply to court asking for a Parental Responsibility Order to be imposed. There is then a requirement in most cases that the applicant to court proceedings sets up a mediation Information & Assessment Meeting with an approved mediator before the court will hear a case.