It's quite easy, provided you comply with all the legal obligations in terms of employees and care of the children.
Whilst teachers in private schools should be CRB checked and their employers and the children's parents satisfied that they are appropriate people to be in charge of children, they do not have to hold recognised teaching qualifications, although most 'mainstream' private schools do (by 'mainstream' I mean those whose aims and philosophy etc are generally the same as those in the state sector).
All the usual rules of setting up a business would apply, because that's basically what it is. A donation-funded school would also need to have some sort of co-operative management strategy.
If your school was 'alternative' in its philosophy, such as teaching mainly, say, farming skills, or where children are allowed to be self-determining, then you would need to state your aims and philosophy clearly to prospective parents.
Privately run schools do not have to be OFSTED inspected, but can choose to be so. Where this is the case, they must comply to OFSTED recommendations. The law only states that children receive an appropriate education. What is appropriate is a matter for those involved, viz. learners, their educators and their parents.