I know of the website for data.gov which provides data but is a publicly available list of taxes paid by publicly own corporations like Apple? If not how do we know how much is paid by them?
For publicly own corporations, you can access their financial statements, where they record the amount of taxes paid (e.g. for Apple here). Notice that where the tax is paid depend on the location of the firm, and whether it is a multinational business. The latter type of firm is particularly problematic, because multinational companies use transfer pricing methods to optimise (i.e. lower) their tax burden. Therefore, it is difficult to determine where those taxes should be paid. Apple and other multinationals seem to be particularly keen to do this (e.g. here).
The reason that sometimes you see headlines stating that big companies do not pay enough taxes locally is because many of these analysis is done by those with access to detailed information like HRMC or Inland Revenue (e.g. here; NB: excuse the doggy source; could not find a better one).
Outsiders to these institutions can have access to this data only in a anonymous fashion (see data available for UK here), so you won't be able to trace Apple's taxation in the UK with such data (or at least you should not be able to do so).