If you go on the tour in England it tells you a lot of information and the creator of it did an amazing job. I know there's been interviews and articles of people actually thinking it was a real bird at first. The reference below pretty much sums it up.
"The Creature Shop team built three different animatronic models of Fawkes; beautiful adult phoenix, an older moulting version and a fledgling newborn that rose from its ashes. Fawkes' plumage was made of real feathers, each individually hand-painted in his fiery phoenix colours. For Fawkes' flying sequences, the Visual Effects team created a digitally animated bird that could perform the soaring, swooping, diving actions that a model could not".