Yes, Donald Trump is obsessed with revenge to his own detriment, often with regard to petty things which should not affect him. It seems that due to his privileged life he is unable to accept criticism and now, as a presidential candidate he is exposed to all sorts of criticism, valid or otherwise, and his thin skin means he spends too much time attacking his critics, even when this harms his standing with the voting public.
Donald Trump clearly hates being slighted and tends to completely lose control whenever it happens. He does this even when the person criticising him is clearly sympathetic to the public, such as the former Miss Universe he humiliated and called a fat pig. His narcissism and his complete lack of empathy for the people he attacks demonstrates that he has a very troubled temperament that would be potentially catastrophic if he was allowed access to real power.
Yes, Donald Trump is obsessed with revenge to his own detriment because whenever Clinton gives him the bait, he takes it. He cannot let others say anything negative about him. He would be egged on as a president by foreign leaders. His skin is too thin to serve as president.
Donald Trump is too focused on revenge, to the point of obsession. This is to his own detriment. He fires off tweets in the middle of the night when he is confronted with what he perceives as a slight. He also made the fight with a Gold Star family very public, drawing unnecessary attention to himself. This is not the behavior of someone that is healthy.