The media spends a great amount of time with Tony Blair and while they may sometimes spin a story to make him look favorable or unfavorable overall they give the public an accurate picture of what the man is like and what his personality is, so overall I think they are pretty accurate.
I do not think there is any reason to believe that the personality that Tony Blair seems to have is being portrayed wrongly by the media. The fact that the image of Tony Blair in the media doesn't seem to be negative makes me think that there shouldn't be anything to argue about.
I think the media accurately portrays Tony Blair's personality. His public persona is no different than a private one. He is a very personable and open person like former US President Clinton. He is very convincing and able to win many people over to his side through his charisma and manners.
Yes, I think that all of the media coverage that has been shown over tony does a good job to get his personality and to get how he is in real life. there is a whole lot off things out there about him, and it gives you a good look.
It is worth noting that any media outlet is most likely influenced by bias and an agenda. It is also true that Tony Blair's public personality, especially in front of the camera is probably quite different than his personal o rprivate life. Which one is the real Tony would be more of a mystery, but a certain act or facade must be established for the cameras to protect Tony's private life.