I am at present working for a private company, can I register a business under self employed or a company?

You will need to check in your employment contract to see whether there's a clause which stops you from doing so. For example, some companies prevents their employees from starting a similar business to them when they leave their job, sometimes for a period of 2 years.

This is necessary as some people will try to get a job in a particular company, learn the trade and get the right contacts and then become a rival to the business.

Therefore it is imperative you scrutinise your employment contract and if need be, you may want to consult a solicitor for professional advise.

PS: Note that I've merged your previous question with this one to make it more complete.