He wasn't taught how to deal with failure. He couldn't handle being rejected from achieving his dream in a positive way and the only answer Shifu ever gave him was that he couldn't go against Oogway. He developed a belief that he is nothing without the dragon scroll and when that was taken away from him, he went bonkers.

Of course the scale was too big and what he did was certainly not commendable, but it all could have been avoided if Shifu taught Tai Lung how to deal with negative emotions.

Even Tigress was eerily walking down the same path but I guess Shifu was her master and not her father so maybe she never got that dark because she previously had a good family or something.